A U.S. citizen does not have the authority to decide whether or not her spouse can remain in the U.S., once that spouse is a lawful resident alien.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
As both an immigration attorney and family law attorney I have to say I've been asked this question many times. I agree with my colleagues above. If the immigrant spouse was granted conditional residency (I.e., the marriage was less than two years old at the time the green card was granted) then they might have difficulty in filing their petition to remove conditions (the I-751 to get the 10 year card) if they're separated.