Your employer cannot require you to produce the marriage certificate unless they can show a compelling reason why the other proofs of your new name are insufficient. Your Social Security card and NYS driver license should suffice.
You certainly don't *have* to give it to them, as you can always quit, or simply refuse (unless you have a contract with them to the contrary). It seems like your question is more directed at whether they can fire you for refusing to provide a copy of the marriage license. In New York, it is illegal to discriminate on the basis of marital status. Here, it seems like they're not inquiring about your marital status, but rather for proof of your marital status. They might be asking for it because they're worried your dodging debt collectors and want to get your paycheck under a fake name so your wages don't get garnished, or for a million other reasons, it's hard to tell. Any other official government document should suffice (i.e. new social security card, or a drivers license with your new last name). The previous poster seems to be familiar with a body of law I am not intimately familiar with that may govern this request, however.
This is a general answer to a general question and is intended for entertainment purposes only. It does not constitute legal advice, and should not be relied upon or regarded as such. It does not create an attorney-client relationship. Only a lawyer with full knowledge of the specific facts of your situation can provide legal advice.