A. Poskim debate whether cooking food in a microwave oven is included in the prohibition of Bishul Akum.
In regards to the prohibition of bishul on Shabbos, Harav Shlomo Zalman Auerbach zt”l (quoted in Shmiras Shabbos Kehilchoso 1:127) maintains that microwave cooking is only rabbinically prohibited, while Harav Moshe Feinstein zt”l (Igros Moshe 3:52) rules that since microwave cooking is a common form of cooking, it is biblically forbidden.
In regards to bishul akum Poskim also disagree. Minchas Chein (1 p. 177) maintains that according to Horav Moshe zt”l food cooked by a Gentile in a microwave oven is considered bishul akum. Shevet Halevi (8:185), Shraga Hameir 6: 52: 3) and Harav Yosef Shalom Elyashiv zt”l (Shvus Yitzchak vol. 6 page 61) assert a similar stringent opinion,
Other Poskim are lenient The arguments for leniency are that when one cooks with a microwave he is not cooking by fire and that microwave technology was not available at the time when Chazal promulgated the Bishul Akum decree and thus was not included in the proscription. Moreover, most food cooked in a microwave oven is not suitable to be served at a king's table or a state dinner.
If the food was already fully cooked by a Jew, all agree that a Gentile may re-heat it in a microwave oven.
Horav Shlomo Miller's Shlit'a opinion is that in case of need such as when caring for the elderly and the ill, one may be lenient.
Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a