The reactions he has gotten from many people who have read his latest book have been encouraging. Apart from his call for a dialogue, the second half of the book is devoted to readings and prayer, though that aspect gets lost in the debate over the book. He had long received desperate messages and impassioned emails seeking counsel or prayer through life’s difficulties, and the publication of “Building A Bridge” prompted even more. He gets about 50 messages daily, in which people talk about things like how a priest would not anoint a dying man in hospice care because he was gay; or how some people were fired from jobs at Catholic institutions because of their sexual orientation.

But he has also been the victim of ad-hominem attacks, even from other Catholics who, he said, do not seem to remember Pope Francis’s remark that “who am I to judge” if a member of the clergy was gay. He tries to ignore the criticism, but when Catholic Vote, a conservative group, sent out a message – which it said was in jest — that Father Martin had been “beaten like a rented mule” by a Dominican priest, he filed a complaint with Twitter that resulted in the account being temporarily suspended.

That, in turn, prompted Austin Ruse, the president of the Center for Family and Human Rights, to call Father Martin’s reaction “pansified,” dubbing him “Father Snowflake” as well as a “perfidious priest.”

Mr. Ruse, who is also a contributor to Breitbart News, did not apologize for using those words. “They are qualifiers for his bad behavior in lying about the comments of Catholic Vote on Twitter,” he said in an email. “I could have used snowflake for he whimpered for a safe space. He lied and caused Catholic Vote to be suspended from Twitter and held up to public ridicule.”

Despite the name-calling, innuendo and canceled speeches, Father Martin said he will press on. He has received support from bishops – who request boxes of his latest book — and from his religious order. He will not step back from social media, saying it is part of his order’s tradition to “find God in all things.”

Yes, even on Twitter.

“We are not afraid of going to the margins,” he said. “That is what Pope Benedict and Pope Francis asked us to do. As Francis said to us, go to the peripheries where the Church has not been serving people or where people need it the most. There is no one more marginalized in the Church than L.G.B.T. Catholics. So, I’m right where I should be.”