Hello,
I'm a green card holder who is been living in the U.S. for the past three years. My boyfriend recently got a tourist visa and came to visit me. After a month, we decided that we want to get married here at a courthouse in the U.S. So, we did that and We have been legally married for a week now and I want to apply for him to get a green card as well. I just want to know if they would allow him to stay and live with me while his green card gets processed or if they would send him back until they give him an appointment at a U.S. embassy.
Thank you.

Normally NO, he will need to return to his nation, and come back when it is approved.

You are permanent resident, not US citizen, so he cannot stay in US while processing the case. Normally it takes approx 3-5 years to get visa number.