I recently read an interview with Rav Dovid Feinstein and he referenced a two year period of Rav Moshe's life that were troubling due to backlash he received following a controversial psak regarding fertility treatments. Does anyone know more to the story? What was the psak? Who disagreed? What was the result?
Rabbi Aaron Rakeffet-Rothkoff has a lecture on YUTorah about Artificial Insemination that discusses it.
Rabbi Feinstein ruled that the Torah prohibition of adultery, and the resulting categorization of the resulting child as mamzer, are only incurred via conventional relations: artificial insemination and in-vitro fertilization are not included. Thus, if a married Jewish man has fertility issues, his wife may be artificially inseminated by some other male donor.
(A similar psak had previously been published by R' Shlomo Zalman Auerbach in the Israeli journal Noam, but that was dismissed as a relatively left-wing publication.)
Yoel Teitelbaum, Grand Rabbi of Satmar, vehemently disagreed. He insisted that any product of a Jewish married woman and another man's sperm constituted adultery, and the offspring would be a mamzer. In his viewpoint, then, Feinstein was allowing adultery and causing more mamzerim to come into the world. Critics of Teitelbaum will point to his hangups concerning sexual matters, which certainly didn't help the situation.
Rabbi Feinstein's family, in the preface to Igros Moshe Volume 8, point to political causes as well: While Rav Moshe was not exactly a card-carrying Zionist, he was involved with Agudas Israel in both the US and Israel, which granted some legitimacy to the State of Israel as it sought to engage it:
רבנו נתמנה אז נם לעטוד בראש אנודת ישראל בארצות הברית. הוא היה יושב ראש הנשיאות שלה, ויושב ראש מועצת גדולי התורה. בשנת תשכ״ד, בא לארץ ישראל לכבוד הכנםיה הגדולה של אנודת ישראל.
בגלל עמדת האנודה לנבי ארץ ישראל וההתננדות חחריפח של חסידות םאטמר לעמדה זו, הוהלט אצלם לתקוף את רבנו ופםקיו(כך ספרו לרבנו אנשי נטורי קרתא בירושלים כאשר היה שם בשנת תשכ״ח. כמטרה להתקפה זו נבהרו תשובותיו בעניין הזרעה מלאכותית ושיעור מהיצה של עזרת נשים. מתלוקת זו שלא היתד, לאמיתה של תורה נרמה לרבנו ענמת נפש מרובה. התקפות אלה לא היו רק באמצעות מאמרים ותשובות, אלא נם בהתקפות אישיות תירופים ונידופים, מעשי אלימות, איומי פצצות, הטרדות טלפוניות ושריפת ספריו.
Our Teacher [Rabbi Feinstein] was appointed as the head of Agudas Yisrael of America. He held the position as its president, and head of its Council of Torah Sages. In 1964 he came to Israel as an honoree of the grand convention of Agudas Yisrael ...
Because of the Agudah's stance concerning Israel and the sharp opposition of Satmar to it, it was decided by Satmar to counter Our Teacher and his rulings. (This is what men of Neturei Karta told him when he visited Jerusalem in 1964.) As an excuse for this conflict, they chose his responsa on artificial insemination and what sort of mechitza was needed for the women's section in synagogue. This conflict, which was not for the sake of the truth of Torah, caused Our Teacher a great deal of anguish. These conflicts were not just in the form of statements or responsa, but vicious personal insults, acts of force, threats of injury, troubling phone calls, and burning of his books.
(Rabbi Rakeffet also tells of a brick thrown through Rav Moshe's window.)
Neither Rav Moshe nor the Satmar ever backed down from their respective halachic stances on the subject; I don't know off-hand exactly when or why the attacks subsided.
(See Bnei Banim #34 concerning honorific titles about a scholar who supports bricks thrown through his opponents' windows.)