Note how this use of the noun form of a verb by itself is similar to, for instance, owari! (Finished; ended).
(The end of the algorithm exercise!)
You could make the -i word a complete sentence like for instance this with
-ni narimasu, which is very humble and polite:
owari ni narimashita
Lit: something has reached the ending state. (Something has ended.)
In the case of 言い過ぎ, since this is abrupt, what is being elided/understood is the coupla だ which would make it a complete sentence:
Lit: It is saying too much. (That is an exaggeration or overstatement!)
Or with more verbiage, adding the topic "sore wa":
References for 言い過ぎ:
So the reason why 言い過ぎ was used rather than 言い過ぎる is because the character wants to say to the other that whatever the other said just now was an exaggeration, rather than to say "you are exaggerating".
Similarly, if the character wanted to say "That is a lie!", he or she might say "uso!" which is a noun by itself or "uso da!". ("usoooooo..." would be more like "no way!" which is different from "You are lying!") To call someone a liar: "uso tsuki!" This is another noun, where a full sentence might be, e.g. "omae ga uso tsuki da yo!".