Regarding whether or not he named the fish, [Rav Saadya Gaon][1] to the verse (viewable [here][2]) writes that Adam named the fish along with all the other animals. This is also stated by  the Midrash Hasserot V'yiterot (printed in Battei Midrashot Vol. II: 14).

However, [Radak][3] to Genesis (2:19) writes that he didn't name the fish, since it would not be feasible to take them out of the water to come be named. [Hizkuni][4] there writes similarly.

[Tosafot][5] to Hulin (66b s.v. kol), however, assumes that Adam only named the fowl and land animals; not the fish, but writes that it is possible that the seemingly superfluous end of the verse stating that "whatever Adam called each living creature was its name" includes fish. 

Notably, the [Baal HaTurim][6] to Leviticus (11:8) suggests that in describing the kosher vs. non-kosher fish, the Torah doesn't use their names since "fish don't have names."

If he did name them, this doesn't address why the Torah wouldn't mention it.


  [1]: https://en.wikipedia.org/wiki/Saadia_Gaon
  [2]: http://mg.alhatorah.org/Full/Bereshit/2.19#e0n7
  [3]: https://en.wikipedia.org/wiki/David_Kimhi
  [4]: https://en.wikipedia.org/wiki/Hezekiah_ben_Manoah
  [5]: https://en.wikipedia.org/wiki/Tosafot
  [6]: https://en.wikipedia.org/wiki/Jacob_ben_Asher