Can an international student in the US marry a green card holder to obtain one?

No. You are free to marry whomever you wish, but it will not convey any status nor will it form the basis for any immigration benefits in the near future, and it will definitely make maintaining your F-1 student status harder.

If you marry someone who already has a green card, you will have unequivocally expressed immigrant intent (making it very difficult to renew or travel on your F-1 visa), but you will not have a marriage that provides any short-term basis on which to immigrate: spouses of lawful permanent residents are in the Family-Based Second preference category (F2A), which is currently backlogged by around 5 years, so a visa number would not be available for you to immigrate until your priority date (filing date of the I-130 petition) becomes current. I recommend that you meet with an immigration attorney to discuss your situation in greater detail.