Of course you can marry your girlfriend, but it will not change your immigration status. While she is not a citizen, she can file a petition for you, under Family Priority Category 2A. That will put you in line, which is about 2 yr 4 mo long at this time. When your turn comes, you have to be in valid status to adjust under this category. If/when she becomes a US citizen, your category changes to that of immediate relative, and your then-current status (or no status) does not matter, as long as you were admitted to the US in any status. Additionally, if you enter as a tourist, your marriage within a short period of time may leave a "bad taste" with immigration for so-called "preconceived intent" to marry, and you may have harder time reentering again.
I agree with everything my colleague says. In addition, I would suggest that your girlfriend consider first become a naturalized U.S. Citizen and then file a Petition for Alien Fiance for you. Hope this helps. Wishing you well.
This communication does not create an attorney client relationship. If I can be of further assistance, please feel free to contact me at any time at either (770)955-1785 or (678)576-9394 or via e-mail at firstname.lastname@example.org. THINK IMMIGRATION - THINK BOB BEER
My colleagues are correct: There is nothing in the law that prohibits you from marrying your Lawful Permanent Resident girlfriend while you are here as a visitor, but that does NOT provide you with an immediate basis upon which you would be able to immigrate or adjust your status, as there is a quota backlog of more than 2 years for spouses of permanent residents (this is the F2A category), nor would you be able to extend lawful stay as a visitor once she files an immigrant petition for you. For any future travel as a visitor, if you get married on this trip but she does not file a petition for you right away, it still creates a presumption of immigrant intent.
This is general information only. It is not intended as a substitute for legal advice, and does not create an attorney-client relationship.