British National Party chief Nick Griffin posted the address of Michael Black and John Morgan after they won their case against a bed and breakfast owner who refused to let them stay in a double room due to religious reasons.
Police said they were investigating the messages on the Twitter feed of the European Parliament lawmaker.
Two of the posts read: "So Messrs Black and Morgan, at (their address). A British Justice team will come up to Huntington and give you a bit of drama by way of reminding you that an English couple's home is their castle. Say No to heterophobia!."
Griffin had earlier made an appeal by Twitter for their address and described the couple as "bullying 'gay' activists".
In an interview with BBC radio, he said discrimination was a "fundamental human right".
"Mr Black and Mr Morgan have the right to decide who enters their home and who doesn't, as do Christians, and that's what they are taking away," he said.
"Those two gentlemen placed themselves in the public eye and they asked for it when they used and abused the legal system to persecute an innocent Christian couple.
"It's time that the silent majority had people standing up for them so that this relentless bullying of ordinary people by activists and the left-wing judiciary stopped."
Griffin insisted he had only wanted a "peaceful" demonstration.
Black, 64, an exams consultant and writer, called the politician an "idiot" and said Griffin "has got very little support in the country".
"Certainly the public response to the incident, when it happened two and a half years ago, and again in the last 24 hours, has been overwhelmingly in support of our stand against discrimination," Black told BBC television.
"So hopefully that will mean the vast majority of the people in the country will just see what an idiot Nick Griffin is and reject his views."
The row came the day after Twitter said it had blocked a neo-Nazi account in Germany in a global first and made another key concession on Friday by agreeing to remove anti-Semitic posts in France.