From your question, it is evident that you are aware that once a food is fully cooked, cooking it further or reheating it is not a violation of bishul (cooking).[1] Your question is whether being fully cooked via a microwave oven is considered cooked. We will begin by examining a precedent regarding nonstandard cooking.
The gemara[2] states that the prohibition of cooking on Shabbat is in effect when something is cooked by heat that emanates from some sort of “fire,” whereas it is permitted to “cook” in the sun. Rashi[3] maintains that the leniency is due to the fact that it is not normal to cook in the sun. The Igrot Moshe[4] asserts that because of the abnormality of sun-cooking, we cannot extend the prohibition of cooking to it from the cases found in the context of the Mishkan, which are the models for what is forbidden on Shabbat. This halacha, however, is not a clear indication that the result of such abnormal cooking is any different from that of conventional methods. Others disagree, based on another gemara,[5] which states that just as there is no Torah-level violation of Shabbat when one cooks something not through fire or its by-product (but, for example, in hot springs), Korban Pesach meat that was cooked in that manner is not subject to the prohibition of eating cooked Korban Pesach meat. Since it is the result, not the process, that is important regarding Korban Pesach, the Avnei Nezer[6] and Minchat Shlomo[7] say that Chazal must have posited that the unusual process produces a result that is different from standard cooked food.
Although the halacha is that there is no prohibition on Shabbat to cook food that was previously cooked, there is a machloket Rishonim regarding whether it is forbidden to cook (in liquid) something that was previously baked (with dry heat) or vice versa. We rule stringently regarding cooking after baking and vice versa, but not in an absolute manner.[8] The Pri Megadim[9] briefly discusses whether it is permitted to put something that has already been cooked in the sun through a conventional cooking process.
The Pri Megadim’s case and that of the Rishonim, cooking something that was already baked, are similar. In both, the food is already fully edible due to a process involving heat before the second process begins. However, the Pri Megadim’s case entails elements of both stringency and leniency in comparison to the case of the Rishonim. In the scenario of cooking/baking, the food already underwent halachic cooking, whereas food that was cooked in the sun did not, such that the Pri Megadim’s case of food that previously underwent nonstandard cooking may be more stringent. But one could also distinguish in the other direction. Cooking something that was previously baked or vice versa leads to a discernable change in the food. That does not seem to happen when one repeats basically the same process, except that the second time is considered halachic cooking. The Pri Megadim’s inclination is that there is certainly no Torah violation to recook food that was previously cooked in the sun, and it is likely permitted, apparently based on the idea that the cooking process is (fully) forbidden only when it changes and/or improves the food.[10]
If we equate microwave cooking to cooking in the sun, your question is equivalent to that of the Pri Megadim. However, that equation deserves evaluation. With the advent of microwaves, the question arose as to whether they produce a new-fangled but essentially standard form of cooking or whether they present a new application of Chazal’s idea that cooking without fire is not considered halachic cooking. (Regarding the practical question of cooking with a microwave on Shabbat, the problems of using electricity essentially preclude it.) The Igrot Moshe[11] claims that since microwave cooking is now a normal form of cooking, its status is derivable from the Mishkan, and it is considered halachic cooking. Although, in context, this leads him to rule stringently and forbid cooking with a microwave on Shabbat as a full prohibition, his presentation implies that microwaving is considered full cooking across the board.[12] Therefore, it should be permitted to reheat microwaved foods.
In fact, even one who would be more stringent than the Pri Megadim regarding cooking items that were already cooked in the sun might agree to permit cooking after microwaving, as the results of microwaving are likely more similar to regular cooking than to cooking in the sun.
The relatively high heat of many of today’s hot plates can make previously cooked rice crispy, and some people like rice like that. Creating such a situation on Shabbat is a problem even for rice that was cooked normally, as this is a significant act of baking after cooking, which is prohibited, as noted above.[13] If one takes steps to make sure the rice will not become crispy, it should be permitted to reheat it on the hot plate on Shabbat.
There is a practical solution for one who wants to be more stringent than what we have spelled out, and especially if he likes crispy rice: He can briefly bake the rice in a regular oven after taking it out of the microwave (five minutes of significant heat should suffice). Then, it will be halachically cooked/baked before Shabbat.[14]