Yes, you can stay in the U.S. and apply for a green card through marriage to a U.S. citizen, as long as your marriage is genuine (not just for immigration).

You’ll need to file Form I-130 (Petition for Alien Relative) and Form I-485 (Application to Adjust Status) these can be filed together since you're married to a U.S. citizen.

 Once you file the green card application, you’ll receive a receipt notice, and eventually, you can also get work authorization (Form I-765) and travel permission (Advance Parole, Form I-131).

This process allows you to adjust your status without leaving the U.S., even if your F-1 visa expires during the process.