I am filing my 2019 taxes and have never filed before, so I expect to get the $1200 stimulus check in the mail. The amount of federal tax I owe for 2019 is less than $1200. Instead of paying the IRS now and getting $1200 later, is there a way to simply have them deduct my taxes owed from the check?
While the system is essentially giving people an advance on a tax credit, trying to have the system account for the 2019 tax return balance would have made it too complex, and slowed everything down.
If you had submitted tax returns for several years, and used direct deposit for refunds, then your stimulus many would have already been transferred. If you had paid by check in the past, then sometime in the next few weeks or months you would receive a paper check, unless you filled out the online form to provide your banking information which might speed up the process. In either of these cases waiting for the check/deposit before submitting your payment is the advice I would give if waiting helps you pay what you owe.
But there is a statement in your question:
I am filing my 2019 taxes and have never filed before
That means that the system has no idea if you will will qualify for the stimulus payment. You may be in the system for your 2018 taxes as a dependent, but unless you have filed in the past they won't send you a payment in the near term. It is possible that you can trigger a stimulus payment soon by submitting your 2019 return. But that would only be true if your 2019 tax situation has you as non-dependent.
But if you could be a dependent on somebody else's 2019 return, you won't see any stimulus payment until the spring of 2021.