A good place to look to find refutations of Christian messianic interpretations of the bible is Sefer Nitzachon, printed in Otzar Vikuchim by Dr. J. D. Eisenstein.
This is his answer to this specific case (p. 256): The Christian claim is that Jesus was crying to G-d, his father, "Why have you abandoned me?" at the time he was being executed. But according to Christian philosophy, Jesus wanted to go down to earth in flesh and blood in order to save all the sinners from hell through his death. If he really wanted to die in order to atone for mankind, why was he crying out why G-d abandoned him? Would he really say "I will call out but I am not answered" if he didn't want or intend to be answered? Furthermore, he refers to "our fathers" (v. 5) — but Christians claim he is the son of G-d, and if so how could he have multiple fathers? Would a representation of G-d on earth call himself a "worm" (v. 7; read תולעת for תועלת in the book)? How can he refer to his brothers (v. 23) if an only son of G-d has no brothers? Why does he praise the descendants of Ya'akov and Yisra'el for honoring G-d (v. 24) if they were the ones killing him, the son of G-d?
This blog post quotes the Radak as challenging the Christian interpretations in pretty much the same words. (This challenge is censored from most current editions of his commentary.) I would guess that the commentary of the Radak was used as a source by the author of Sefer Nitzachon.
To add just one Jewish interpretation of the psalm, the Targum (online here, apparently translated into English here) of an uncertain date reads it, like Rashi, as referring to the state of Israel among the nations. Other answers have already preceded this one, and better address Jewish interpretations of the Psalm, whereas this one focuses more on polemics against the Christian interpretation.