The best way you could use it as a standalone word would be,
J'ai mangé (toute) la tarte et, depuis, j'ai mal au ventre.
As said before. The reason is simple: the commas are used to do what we call a "proposition explicative" in french, therefore the sentence could be valid and have the same meaning without it.
J'ai mangé (toute) la tarte et j'ai mal au ventre.
"Depuis" is here used to make explicit a cause-and-effect principle: that's its usage in every sentence.
If we want to push the analysis further, here is how you could construct this sentence to understand it:
First, you have two sentences:
J'ai mangé (toute) la tarte.
J'ai mal au ventre.
Then you use "et" as a preposition, to join the two sentences:
J'ai mangé (toute) la tarte et j'ai mal au ventre.
Note that you could use just a comma, instead of the "et".
This sentence is correct, but it does not imply any cause-and-effect relation between the fact that I ate the whole pie and that now my stomach aches. Here comes "depuis", and your sentence now has its full meaning.