One word for “saying ‘Farewell’ to someone”

Dismiss and usher out come close to fulfilling that role.

dismiss: to direct or allow to leave

usher out: end one's encounter with someone by causing or permitting the person to leave

Alternately, salute can also be used as an expression of farewell.

salute: to greet or address with an expression of welcome, goodwill, or respect

He saluted me and left.

Since the headline formulation of the question asks for the single-word equivalent of a gerund phrase, I will venture a noun: valediction. Per OED the verb form valedict is “rare.”

"Goodbyes" is a single word noun that can be used as you want. So is Farewells

I said my goodbyes to my school friends at graduation.

Parting(s/words) work here.

done or said by someone when they are leaving

And adieu is understood but is similar to saying your goodbyes.

a farewell remark

I bid adieu to all of my friends before leaving.

Say your adieus before leaving the function.

Not in particularly common use, but there is a present participle of farewell: "farewelling".

See, for example, "Farewelling troops at Lyttelton, 1940".

Um, the single word for "say farewell" is farewell. This is English. You can use any word at all as a verb.

• Wiktionary:

  farewell (third-person singular simple present farewells, present participle farewelling, simple past and past participle farewelled)

  1. To bid farewell or say goodbye

     • 2009 February 9, Neil Wilson and staff writers, “Tributes for newsman Brian Naylor and wife, killed in fires”, Herald Sun:
       He farewelled viewers with a warm sign-off after each bulletin: "May your news be good news, and goodnight."

• Merriam-Webster:

  transitive verb
  chiefly Australian & New Zealand
  : to bid farewell to
  The retiring teacher was farewelled by the whole school at a special assembly.
  First known use: 1580

As I commented, "seeing off" is another example in fairly common modern usage, and in a similar manner to your use of the word 'greeting'.

It also allows a range of tenses in the normal conjugation of 'to see':

"I saw them off as they left for their holiday"

"Before seeing them off, I gave their daughter a gift for her upcoming birthday"

"Roger will see you off"

And many others. As you can see these are readily replaced by "greet", "greeting", and "greeted".

The only verb I can think of with any real pedigree is valedict, which describes the act of bidding goodbye just as salute describes that of bidding greeting. The Latin etymology of the two words is helpful: salve(te) is a Latin imperative used to greet someone, from the verb salveo (imperative because it really means "be well"); from this verb came the verb salutare, to bid greeting, whence the substantive salutatio, a greeting. Conversely, to bid farewell, one said vale(te), another imperative (meaning something like "be strong"); hence the verb valedico, to bid farewell, and thence the substantive valedictio, a bidding farewell.

From these two Latin verbs come the names of the speeches given at graduation ceremonies in the West -- one to bid greeting (the salutatory speech), and one to bid farewell (the valedictory speech). And thence the common verb "to salute" (and noun "salutation"), as well as the much less common verb "to valedict" (and the somewhat commoner noun "valediction").

I vote we mount a concerted effort to bring "valedict" more into vogue.