Marriage itself doesn't violate the visa, but the intent to remain permanently in the U.S. does.
And no, you cannot apply for permanent residency through marriage unless (1) you return to your home country for the requisite two years, or (2) you obtain a waiver of the residency requirement (which is borderline impossible on some J's, like the Fulbright). I strongly encourage you to meet privately with an immigration attorney before you do anything so that you're sure to have accurate information. Do not take action on the basis of self-help or internet research.
THIS ANSWER IS FOR INFORMATIONAL PURPOSES ONLY, DOES NOT CONSTITUTE LEGAL ADVICE, AND DOES NOT CREATE AN ATTORNEY-CLIENT RELATIONSHIP. Immigration is complicated and the wrong action can have serious consequences. Never take action in your case based solely on general information like is offered here. Speak to an attorney who can give you specific advice about your own situation.
I agree with my colleague.
This answer is of a general nature and should not be relied upon as final, nor is it intended as legal advice. Consult with a qualified attorney before making any legal decisions. Gen Kimura, (832) 247-6932.
I agree with Ms. Allen.
Behar Intl. Counsel 619.234.5962 Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.