Refusing to sign does not stop a divorce in New York. It changes the route the case takes and usually makes it slower, but one person cannot hold a marriage together by ignoring the mail.
Why refusal does not block it
New York allows a no-fault divorce on a sworn statement that the relationship has broken down irretrievably for at least six months. That statement comes from the person filing. The other spouse's agreement is not what the divorce depends on.
What happens instead
Once served, the other person has a limited window to respond. If they do not respond at all, the case can proceed on default, and the judgment can be granted without their participation.
If they respond and disagree about property, support or children, the case becomes contested and moves onto a different track with court appearances.
The practical difference
Not signing rarely changes the outcome of whether the marriage ends. It usually changes the cost and how long it takes, and it removes that person's input into terms a judge will then decide.
This page is general information about how the process works in New York. It is not legal advice and it does not create an attorney-client relationship. Whether any of it applies to you depends on your situation, which is what a consultation is for.