I had to read your post 3-4 times to fully understand what you were asking, but I think I got it now.
You never mentioned whether this was IRA or 401K, so I'm going to assume it's 401K with a limit of $17,500. If it was an IRA you are ignoring the rules (i.e.-the $5,500/$6,500 annual IRA limit). The simple answer to your question is yes, sort of.
Let me re-phrase the question based on my understanding of it: If I have 0 kids today, but theoretically (or actually) have 1 kid at the end of 2014, will this child tax credit plus a contribution to a ROTH allow me to pay the same amount of tax as if I had 0 kids and an equal contribution to a TRADITIONAL? If that is your question, and again ignoring the IRA limits, the answer to your question is yes. Here's an actual example of how it would play out based on 401K rules:
0 Kids 1 Kid
Traditional Roth
401K 401K
Taxable Income in your Example $24,816 $24,816
Additional Exemption - (3,950)
Add Roth Contribution - 10,616
Taxable Income 24,816 31,482
Tax 2,815 3,815
Child Tax Credit - (1,000)
Total Tax 2,815 2,815
I added the Roth contribution because if you're taking traditional contributions and turning them into Roth, this will increase your taxable income. I increased your $6,666 contribution to $10,616 because of the $3,950 exemption as Christiana pointed out. So, if you are currently contributing to a traditional 401K, and you want to change to a Roth because you are having a kid and you want to keep your tax bill the same, you have up to $10,616 in "tax-free Roth contributions" as you stated in the original post.
I did not factor in the savers credit, or EIC, or anything else as it gets too convoluted. You also specifically stated taxable income, so I used that as my baseline for the scenario. If you meant gross W-2 income, net W-2 income, AGI, or something else these numbers aren't good. I hope that helps. If not, let me know and I'll re-run with your actual intentions/numbers.