Shouldn't it be "See you on Monday." instead?

Before leaving, the woman also told her superior, “See you on Monday”.

Shouldn’t it be “See you on Monday.” instead? If not, why is it so?

Thanks.

You are correct.

See you on Monday.
This is an incomplete sentence (sentence fragment).

Sentence fragments usually require an upper case letter at the start, and a period at the end.

This is also a quotation. As a quotation at the end of the sentence, the period belongs inside the quotation mark.