Can I avoid paying capital gains tax if my income is below a certain amount?

Yes. The criterion is taxable income. 
https://www.irs.gov/taxtopics/tc409#:~:text=Capital%20gains%20tax%20rates,than%2015%25%20for%20most%20individuals.

But remember that "taxable income" includes the capital gain itself.

Avoid?  Good wording.