How long has it been on your fruit? A week is a bit short to extract all the potential flavors.
The tartness is an acquired taste, loved by some hated by others. I seriously doubt lime (Calcium Hydroxide, Ca(OH)2) will improve it, and most likely will make it totally undrinkable. Yeah, it may counteract the acids chemically, but the residual flavor may be way worse.
If you need to test ad hoc additions, I would add very, very small amounts of lime to a premeasured amount of beer, and taste the results. Then decide where you want to go with it.
As said before, you definitely need a hydrometer to accurately measure the remaining sugar. You're fermenting blind at the moment.
I wish you had an accurate FG reading before you added the fruit. That would have been your base. It's not uncommon for fruit additions to sour your beer. Depending on its source it may have added lacto bacillus and/or other bugs to your beer.
If you don't like tart beers, there maybe ways to back sweeten it, but I have no experience with that, particularly if you have bacteria in there. They're much harder to stop/kill than yeast, and they can eat through practically everything organic.
I actually like the sours.