Can I marry a woman who's visiting on a tourist visa?

Although not mentioned in your question, I assume you are a U.S. Citizen. Generally, a U.S. Citizen may marry someone who entered the U.S. with a visitor's visa, and then sponsor her in the "Immediate Relative" visa category so the foreign national spouse may "adjust status" to become a Lawful Permanent Resident (to get a "Green Card"). If the marriage takes place and the petitioning begins fairly soon after the foreign national entered the U.S. with a visitor's visa, however, then the USCIS may suspect "visa fraud" - entering with a visa that is exclusively for people who intend to stay temporarily and then return abroad, while actually intending to become married and stay permanently. It would be wise to engage an immigration attorney to learn all of the relevant information about the foreign national and the plans to marry, in order to be able to advise about eligibilities, options and strategies, and to assure that the adjustment of status process is handled properly and is fully documented. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.