He was educated at the Rabbi Jacob Joseph School, the Talmudical Yeshiva of Philadelphia, the Telshe Yeshiva in Cleveland, and then at Yeshivas Torah Ore in Jerusalem, under the leadership of Rabbi Chaim Pinchus Scheinberg. He received four semichas: from Rabbi Serayah Deblitzky, a student of the Chazon Ish; Rabbi Ovadia Yosef, former Sephardi chief rabbi of Israel; Rabbi Betzalel Zolty, Ashkenazi chief rabbi of Jerusalem; and Rabbi Elazar Shach, a rosh yeshiva at the Ponevezh Yeshiva in Israel.
After leaving Torah Ore, Tropper accepted a position at the nascent Ohr Somayach institutions. After five years in Jerusalem teaching baalei teshuva, Tropper returned to the United States to become the educational director of Ohr Sameach, New York located in Yonkers and Monsey. In 1981, Tropper joined Rabbi Avrohom Gershon Tress zt"l in founding a new yeshiva for baalei teshuva, the Kol Yaakov Torah Center, and became its rosh yeshivah.
In 1995, Tropper founded a new outreach program called Horizons, focusing on educating and inspiring the unaffiliated to experience a Torah lifestyle. As its director, Tropper traveled the world lecturing and teaching. He has also written three books: a halachic work based on the decisions of his mentor Rabbi Scheinberg, a guide to the laws of yichud, a discussion of the prohibition of seclusion between men and women.
Tropper founded the Eternal Jewish Family (EJF) project, which advocates more rigorous standards for conversion to Judaism, and assists intermarried families with conversion issues. Until 2009 he was the director and chairman of EJF's Rabbinic Committee. The organization was funded by billionaire Thomas Kaplan and Kaplan's nephew, Guma Aguiar, who donated at least $8 million to the project, according to IRS documents. In October 2009, Tropper was sued by Guma Aguiar for allegedly misappropriating funds given to him by Guma for charitable purposes. Tropper says that Aguiar threatened to throw him out a window. Aguiar's attorney denied any threat was made.
In 2006, The Jewish Week reported allegations that Tropper revoked a woman's conversion after learning that the woman "occasionally wore pants". Tropper denied this accusation, claiming the woman had in fact been breaking the Shabbat after her conversion, and that his role was confined to relaying this information to the appropriate authorities.
In early December 2009, recordings circulated on the internet that contained conversations between Tropper and a woman whose conversion he had been supervising, encouraging the woman to engage in sexual activities with others; the two also discuss having sex with each other. Tropper released a statement through his attorney that admitted no wrongdoing and expressed regret for “what has appeared to be conduct not within our significant laws of modesty.”
On December 14, 2009, the EJF announced that Tropper had resigned two days earlier, that he had been replaced as chairman of the Rabbinic Committee by Rabbi Elya Ber Wachtfogel, that Rabbi Chaim Blum would be taking over as interim director, and that Rabbi Reuven Feinstein, the chairman of the Halachic Committee, would take a more active role in overseeing the organization's activities. No reason was given for these changes. On December 18, 2009 Wachtfogel issued a letter denying that EJF is under his direction. The press release was then deleted from EJF's website. On December 20, EJF put out a new press release omitting all mention of Wachtfogel or the chairmanship of the Rabbinic Committee.
However the New York Post reported of tapes in which Tropper is heard encouraging a pretty, blond 32-year-old woman who Tropper was guiding through conversion to participate in phone sex and actual sex with men the rabbi knows, including one he calls "the Satmar guy." Tropper, who calls the woman "darling" and "cutie pie," talks about his own love affair with her at one point, saying: "I want to squeeze you." He also fantasized about rape. "I could role-play a rape with you but I couldn't actually rape you -- you know what I'm saying, darling -- does that make sense?" he asks. The scandal has rocked the Orthodox community from Rockland to Israel, and transcripts of the sex tapes are circulating on Jewish blogs, with the audio posted on YouTube. The rabbi talks about paying her money for a lawyer and a stipend of $1,300 for the month of November. He also mentions putting in writing an agreement between the two. "Why would you want to document that kind of agreement on an e-mail?" she asks. The woman, who identifies herself on the tape, apparently recorded the phone conversations. "It was only supposed to go to a few leading rabbis," She later told The Post. She refused to comment further, and later released a statement saying, "While an individual 'rabbi' acted in an inappropriate manner, my desire to become a bona fide Jew is undeterred." She also reportedly told a Jewish blog, failedmessiah.com, that Tropper would tell her: "If you fulfill my needs, I'll fulfill yours -- and you need a conversion."
On 14 February 2010 Tropper was to be summoned to a meeting in the Yeshiva Bais Mikroh of Monsey where the Rabbonim were to demand that Tropper resign all his posts and leave Monsey. Prior to the meeting, Tropper signed an agreement which stated “In not fulfilling the requirement of Shulchan Aruch that a leader be 'Pirko No’eh' (פרקו נאה), the undersigned is relinquishing his position.” Tropper also stated that he "released all rights and severs any connection he has to his former Yeshiva in Monsey and withdraws forthwith as a trustee of the organization." Tropper also stated his agreement to move out of Monsey within a few months.