Moshe refrained from marital relations with Tzipora for almost forty years, so as to be able to receive prophecy at any moment (Shabas 87:1). 1. Did he divorce her? 2. If not, why not?
Rashi (to Num. 12:1) says ועתה גרשה - now (around the time it came to Miriam's attention) he had divorced Tzipporah.
The Lubavitcher Rebbe points out (Likkutei Sichos, vol. 18, p. 145, marginal note to footnote 41) that Rashi gets this from the fact that the Torah calls her האשה, "the woman," rather than אשתו, "his wife." He also notes that according to Rashi, then, there were two stages in Moshe's withdrawal from Tzipporah: first he ceased marital relations with her, then later (perhaps because of the halachic issues involved - Ex. 21:10 and Shulchan Aruch, Even Ha'ezer 76:11) he divorced her.
Moshe told Miriam that he would not divorce his wife for precisely for the factors that Miriam was calling to his attention. "When I was a fugitive and I was a poor penniless shepherd, this woman married me. She stuck by me when I was a nobody. Now that I am the 'Gadol HaDor,' the teacher of all Israel, and the master of all prophets, I will not abandon her.
Ran (D'rashos 8), in discussing the reason why Miriam and Aharon did not speak disparagingly of Moshe until then, writes that before then, they had thought that Moshe seperated from his wife because he was too busy with his leadership position and its duties. Now that he had gathered 70 z'kenim to assist him with the leadership and he was still separated from his wife, they understood that it was really because of his prophesy.
It is clear, therefore (since the Ran assumes that Miriam and Aharon expected Moshe to resume his original relationship with his wife after appointing the elders), that according to Ran, Moshe had merely ceased his conjugal relationship with his wife, but had not actually divorced her.