The child tax credit is phased out based on your total income (Modified Adjusted Gross Income, or MAGI). The type of income doesn't matter. For this purpose, net capital gains are the same as any other type of income. The credit begins to phase out if your income is more than $200,000 ($400,000 if married filing jointly). The credit is completely eliminated if your income is more than $239,000 ($439,000 if married filing jointly).
So there is no specific limit for capital gains, but capital gains could affect your eligibility for the child tax credit if adding capital gains to your other income makes your total income more than the phaseout threshold.