I cannot fret with it. So I am learning as a lefty. I also cannot hold the pick in this hand. So is it ok to just use the fingers which I can do? I am a beginner.
Take no notice. Press on as you are. There have been plenty of remarkable guitar players who have found stranger ways to play. You sound like you want to play despite having found problems doing it in the 'conventional' way. Good for you. People rest the guitar on their laps and play that way (which I feel is actually more productive, once one's accustomed),so just get on with it, and make the guitar make music however you play it. There's actually no reason for using a pick, if you can play without, fine!!
As an afterthought, I play with several players who play without plectrum, and each time I think I wish I could ditch mine!
Fingerstyle guitar is awesome, but it's hard to say whether that would be a good fit for you without more detail — I would guess that you don't have the strength or control.
However, if you're thinking of playing with your fingernails acting as a pick you can definitely do that!
A number of well-known guitarist just use their fingernails, frequently just growing out the thumbnail (which might work better for you since the thumb has the strongest muscle). You can also curl your fingers and use the backs of the nails to strum, which has a neat effect with multiple strikes on the strings.
You might want to get some clear nail polish if you are playing consistently, to prevent your nails from wearing through or chipping.