Why wasn't Harry punished after he used Sectumsempra on Malfoy?

Well, could be that Draco instigated the fight first with an Unforgivable curse. He cast a Cruciatus curse that missed Harry, and Harry, defending himself, cast the only spell he could think to remember in that heated moment. They both would've been expelled (probably) had Snape reported the incident, but that would mean Harry was out of Hogwarts (safest place to be) and Draco couldn't follow through with Voldemort's assignment to kill Dumbledore.

Just guessing.

It is a dark curse, and a dangerous one at that, but it is the same scenario in both the book and the movie. This is an unknown spell invented by Severus Snape when he was younger. It is an unknown curse to the public and Snape tried to keep it quiet by just giving Harry detention. If it was a known curse, it may have been an Unforgivable Curse and Harry may have been expelled or arrested.

The reason that harry wasnt expelled is because Snape made the spell which dumbldore would be able to tell from the book which harry would have said that he found the spell from. Or that he cared about Lily too much and that makes Snape protect harry like he always has before.

Harry Potter was punished if you would read the book. He got detention with Snape and in a way, seeing what it done to Draco Malfoy was also basically a punishment in itself. He was horrified at what he had done and he felt guilty because he couldn't fix it. In other words, he got a mental punishment from Snape (making him copy the names of Sirius Black, Reamus Lupin, James Potter and Peter Pettigrew) a physical punishment from Snape copy old occurrences and a moral punishment from seeing what he had done to Draco Malfoy. I rest my case.

ETA 4.17.14: We've spent so much time going back and forth on what constitutes an acceptable answer, and which citations are acceptable, I'm afraid I overlooked an integral part of your question. You write, Harry used a dark curse on Malfoy (in the 6th movie). I missed that you were asking about the movie version of Sectumsempra, not the book version. In the movie version, the book version is completely removed; we do not see Harry receive any punishment for hurting Draco, as you note, nor does Dumbledore bring it up or lecture Harry about it.

The scene was changed quite drastically. Per TV Tropes, the film makers use the aftermath of Sectumsempra to alter Harry's motivation for hiding the Half-Blood Prince's book in the Room of Requirement. In the book, Harry hides the book for the selfish motivation of not wanting Snape to confiscate it. In the book, Harry really shows little feelings toward Malfoy and his friends reinforce this by stating it seemed as if Malfoy was going to cast the Cruciatus Curse, which justified Harry's actions. In the movie, Harry's motivation for hiding the book is far more contrite -- he hides the book so he can't "be tempted" (Ginny's words) to use Sectumsempra again, or explore any other dark spells that might be in the Half-Blood Prince's copy of Advanced Potions Making.

Even though you marked the above answer as correct, I notice that Harry's exact punishment for casting Sectumsempra against Draco is not detailed as it appears in canon. As you may know, I believe a direct canon quote is a more sound and reliable citation than a link to, and a C&P from, the Wikia. Of course, you are absolutely free to disagree with me. :) Anyhow, onward! Harry and Sectumsempra:

Yes, Harry was punished for Sectumsempra. The 1996-1997 timeline at the Harry Potter Lexicon puts the Sectumsempra incident "circa May", but does not specify an exact date in May of 1997. Regarding punishment, he received detention with Professor Snape every Saturday, starting from the time of the incident with Draco until the end of term. (I apologize in advance for the number of quotes from Half-Blood Prince, but they are all relevant and specific to the question, aside from the last one, which is a quick and (I think) amusing read.

‘Do you know what I think, Potter?’ said Snape, very quietly. ‘I think that you are a liar and a cheat and that you deserve detention with me every Saturday until the end of term. What do you think, Potter?’

‘I – I don’t agree, sir,’ said Harry, still refusing to look into Snape’s eyes.

‘Well, we shall see how you feel after your detentions,’ said Snape. ‘Ten o’clock Saturday morning, Potter. My office.’

‘But, sir ...’ said Harry, looking up desperately. ‘Quidditch ... the last match of the –’

‘Ten o’clock,’ whispered Snape, with a smile that showed his yellow teeth. ‘Poor Gryffindor ... fourth place this year, I fear ...’

_{Half-Blood Prince - page 494 - Bloomsbury - chapter twenty-four, Sectumsempra}

Harry misses the last Gryffindor Quidditch match of the season, against Ravenclaw, wherein Gryffindor wins the Quidditch Cup, because he had to do detention with Snape, where Snape makes Harry copy by hand Filch's old records of prior wrongdoers and the punishments they received; specifically, Snape had Harry re-do the punishment cards for his father, James Potter, Sirius Black, Remus Lupin, and Peter Pettigrew:

‘I thought you could start,’ said Snape, a malicious smile on his lips, ‘with boxes one thousand and twelve to one thousand and fifty-six. You will find some familiar names in there, gvwhich should add interest to the task. Here, you see ...’

He pulled out a card from one of the topmost boxes with a flourish and read, ‘“James Potter and Sirius Black. Apprehended using an illegal hex upon Bertram Aubrey. Aubrey’s head twice normal size. Double detention.”’ Snape sneered. ‘It must be such a comfort to think that, though they are gone, a record of their great achievements remains ...’

[...]

It was, as Harry had anticipated, useless, boring work, punctuated (as Snape had clearly planned) with the regular jolt in the stomach that meant he had just read his father or Sirius’s names, usually coupled together in various petty misdeeds, occasionally accompanied by those of Remus Lupin and Peter Pettigrew.

_{Half-Blood Prince - pages 497-498 - Bloomsbury - chapter twenty-four, Sectumsempra}

Note that Professor McGonagall was so angry at what Harry did to Draco that she maintained Harry was lucky to have not been expelled from Hogwarts:

Harry had already been called out of the common room to endure fifteen highly unpleasant minutes in the company of Professor McGonagall, who had told him he was lucky not to have been expelled and that she supported whole-heartedly Snape’s punishment of detention every Saturday until the end of term.

_{Half-Blood Prince -- page 495 - Bloomsbury - chapter twenty-four, Sectumsempra}

Frankly, I was surprised that Harry received only detention after hitting Draco with Sectumsempra -- Malfoy could have bled to death or had permanent scarring on his face and chest.

‘SECTUMSEMPRA!’ bellowed Harry from the floor, waving his wand wildly.

Blood spurted from Malfoy’s face and chest as though he had been slashed with an invisible sword. He staggered backwards and collapsed on to the waterlogged floor with a great splash, his wand falling from his limp right hand.

‘No –’ gasped Harry.

Slipping and staggering, Harry got to his feet and plunged towards Malfoy, whose face was now shining scarlet, his white hands scrabbling at his blood-soaked chest.

_{Half-Blood Prince -- page 489 - Bloomsbury - chapter twenty-four, Sectumsempra}

The Sectumsempra chapter is important for three additional reasons:

1. After Harry leaves detention and returns to Gryffindor Tower, he learns that Gryffindor has won the 1997 Quidditch Cup. He swoops Ginny up into a big ol' kiss, thus establishing the Harry/Ginny ship as canon.
2. The Sectumsempra incident may be the only time in the entire series where Pansy Parkinson might have actually been justified in harping on Harry -- Sectumsempra is a dark and dangerous spell -- could it be fatal? I think it possibly could be if the person attacked were then left alone with their injuries -- they could potentially bleed to death.
3. It is in the Sectumsempra chapter that Harry borrows Ron's copy of Advanced Potions Making to switch out for the Half-Blood Prince's copy, so Snape won't find out Harry has it. Ron has written his name on the inside cover with one of Fred and George's Spell-Checking quills ... and apparently didn't check the results:

‘This is the copy of Advanced Potion-Making that you purchased from Flourish and Blotts?’

‘Yes,’ said Harry firmly.

‘Then why,’ asked Snape, ‘does it have the name “Roonil Wazlib” written inside the front cover?’

Harry’s heart missed a beat.

‘That’s my nickname,’ he said.

‘Your nickname,’ repeated Snape.

‘Yeah ... that’s what my friends call me,’ said Harry.

‘I understand what a nickname is,’ said Snape.

_{Half-Blood Prince - pages 493-494 - Bloomsbury - chapter twenty-four, Sectumsempra}

Harry did not intentionally mean to hurt Malfoy to that point, when he acquired the book and saw the notes for the spell said it was for enemies. Thus he decided to try to use it on Malfoy. He was very guilt ridden and horrified by what he had done, and yes he definitely did get in trouble.

Source : http://harrypotter.wikia.com/wiki/Sectumsempra

Many years later, in the 1996–1997 school year, Harry Potter came into the possession of Snape's textbook, and learned the incantation. Without knowing its effects at the time and believing that the "Half-Blood Prince" (Snape's nickname signed into the book) merely copied it as a note of reference, Harry became interested in trying it, thinking it would have some hilarious effects as with Levicorpus and other spells found within the same book. He considered using it against Cormac McLaggen for annoying him, though it was not until he confronted Draco Malfoy in Moaning Myrtle's Bathroom that he utilised it, mortally wounding Malfoy. Snape saved Draco, and having realised that Harry got a hold of the old textbook, he punished Harry with a multitude of detentions for nearly killing Malfoy. Harry, despite disliking Malfoy, did not truly want to harm Malfoy to such an extent, and was both horrified and guilt-ridden by using the curse against him. Professor McGonagall considered Harry to be lucky not to have been expelled for such an act, and she full-heartedly supported the detentions, while Pansy Parkinson wasted no time in vilifying Harry.

Harry Potter and the Half-Blood Prince (Chapter Twenty-Four - Sectumsempra)