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").