The Torah prohibition against partaking of meat with milk does not apply to eating fish with milk; fish does not qualify as "meat" with regard to this prohibition. Strictly speaking, then, Halacha does not forbid eating fish together with milk or other dairy products, neither on the level of Torah law nor on the level of Rabbinic enactment.
However, Maran (author of the Shulhan Aruch), in his magnum opus, Bet Yosef (Yore De’a section), writes that one should not eat fish with milk because this combination poses a health hazard. He references in this context his comments elsewhere in Yore De’a (173) describing the health hazard posed by eating meat together with fish. It appears that Maran compared the two situations of fish with meat and fish with milk, viewing both as a potential risk to one’s health which must be avoided.
Some later writers, however, contended that this passage in the Bet Yosef is the result of a scribal error. According to this theory, Maran actually wrote in this passage about the hazards posed by eating fish with meat, not by eating fish with milk. These Poskim thus maintain that there is no problem whatsoever with partaking of milk with fish. Several leading Ashkenazic Halachic authorities follow this view, including the Rama (Rabbi Moshe Isserles of Cracow, 1525-1572), the Shach (Rabbi Shabtai Hakohen, 1621-1662) and the Taz (Rabbi David Halevi Segal, 1586-1667). For this reason, Ashkenazim are generally lenient with regard to this issue, and permit eating fish with milk or other dairy products without any concern. Thus, for example, it would be permissible for Ashkenazim to eat lox with cream cheese, or a tuna malt.
Among Sepharadim, however, different views were taken by the Halachic authorities. The Hida (Rav Haim Yosef David Azulai, 1724-1806) followed those opinions that attributed the comments in the Bet Yosef to a scribal error, and thus permitted eating fish with milk. Others, however, including the Knesset Hagedola (Rabbi Haim Banbenishti of Turkey, 1603-1673), held that the prevalent text of the Bet Yosef is valid and authoritative, and one must therefore refrain from eating milk with fish. In fact, Rabbenu Bahye Ben Asher (Spain, 1263-1340), a scholar from the Medieval period, writes explicitly in his Torah commentary (Parashat Mishpatim) that partaking of fish with milk poses a potential health risk. This is also the opinion accepted by the Ben Ish Hai (Rabbi Yosef Haim of Baghdad, 1833-1909), Hacham Baruch Ben Haim and Hacham Ovadia Yosef. Additionally, Hacham Yom Tob Yedid, former Chief Rabbi of Halab, reported that the practice among the Jewish communities in Syria was to follow the stringent position and refrain from eating fish with milk.
As for the final Halacha, those who act leniently in this regard certainly have authorities on whom to rely. However, in light of the stringent position taken by Hacham Ovadia Yosef, whose rulings we generally follow, and given that this was the practice of our forebears in Halab, it would seem preferable to avoid eating fish together with milk or other dairy products. It should be noted that there is greater room for leniency regarding fish with butter, which apparently is not seen as posing a health risk. Therefore, even those who follow the stringent position may eat fish with butter sauce, as is served in some restaurants.
Summary: Different views exist as to whether one may eat fish with milk or other dairy products. In practice, Ashkenazim act leniently in this regard, whereas different customs exist among Sepharadim. The custom in Halab was to avoid eating fish with milk, and this is the ruling as well of Hacham Ovadia Yosef. Even those who follow the stringent position may eat fish with butter.