My Virginia state tax was Accepted back in March, but the payment I owe the state has not yet been deducted from my bank account. Should I be worried?

Unless you picked a different date the default would be 5/1...

Hmm. Maybe that is my situation. I know for Fed I picked an earlier date, and I thought TurboTax might have carried that same date. Maybe they just went with the 1st. Guess I'll find out tomorrow. 🙂

did you specify a date for withdrawal, if so has it been more than 2 weeks past that date.  if so or no date specified, call them you got nothing to loose

https://tax.virginia.gov/contact-us

Thanks. Will give them a call if money doesn't get pulled tomorrow.

Was your state return accepted?  Better double check.
https://turbotax.intuit.com/tax-tools/efile-status-lookup/

Then Check the printout or PDF of your return; look for the state cover sheet with the Turbotax logo.  If you owed tax, it will show the payment information and how/when you decided to pay.  Read the state payment instructions carefully since most states cannot be paid from within the TT system and requires additional steps.
https://ttlc.intuit.com/questions/1900884-what-are-my-options-for-getting-my-state-refund-or-for-paying-any-state-taxes-i-owe

 To get whole return with all  the worksheets please log back in your TurboTax account and "add a state" to open the return back up (you are not really adding a state, just to open it back up), then select Tools, Print Center and choose the print center, then select returns and choose the radio button to include all worksheets.

I believe that VA is one of the states that does NOT let you pay this way.

I've been doing my VA taxes via TurboTax for as long as I can remember. Hopefully I just forgot to set a custom date for payment and it happens by tomorrow.