On Passage, Objections to the Contrary Notwithstanding

This is legal language

On this site I found this sentence which made me understand the expression better:

The question is, Will the House on reconsideration agree to pass the bill, the objections of the President to the contrary notwithstanding?

which means that the bill may be passed despite the objections of the President not to pass it (to do the contrary).

So I would understand your expression in this way:

On Passage (of some law or bill), (in spite of) Objections to the Contrary (Notwithstanding)

Bottomline, this law was passed in spite of all objection not to pass it.

Here is an article about the ambiguity of notwithstanding used in such constructions.

This question might also enlighten you more (it explains another similar structure: "myths to the contrary notwithstanding").

As Xanne has pointed out in the comments, this a set phrase regularly used in the documents of the U.S. Congress. Its intended meaning is:

objections of the president[, which present arguments] to the contrary[,] notwithstanding.

What makes the phrase objections to the contrary confusing is that objection to X, normally means that the objection consists of some arguments against X. Consequently, objection to the contrary of X, taken at face value, would mean something consisting of arguments against a position that is contrary to X, which would amount to supporting X (the two negations cancelling out). This is clearly not what was intended here (if it had been, then the presence of notwithstanding would be puzzling).