I am a graduate student from India. Can I bring my girlfriend here in the US from India to get married with her?

You need to comply with Florida's marriage laws to get married here. I don't know if your marriage would be recognized in India. You should check with a lawyer there.

Each country has its own laws. And in the U.S. each of the states have their own laws regarding family law, i.e. marriage is state issue, not a federal issue. In any case, what another country is willing to accept, adopt, ratify, or recognize is up to that country's law. Therefore, I recommend that you speaking with an attorney in India, to help address your concerns. On a completely different matter, but very relevant in your circumstances, you should consider speaking with a U.S. immigration attorney to discuss the process for obtaining the appropriate documents, visas, etc., and to learn about the time-frames and requirements needed to obtain U.S. citizenship if that is something that you/she is considering.

You need to talk to a Family Lawyer asap.