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)? 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.)
As for Jewish interpretations of this chapter, other answers have already preceded me. Targum Yonasan (online here, apparently translated into English here) is the oldest written rabbinic interpretation we have on it (c. 450 B.C. according to linked site). He reads it pretty much the same way as Rashi.
*There is a misprint in Otzar Vikuchim where he substitutes תועלת for תולעת. This is an obvious misprint, and תולעת is the version in the actual verse.