My Federal (and State) returns were accepted on May 1. It's June 2nd, now. I owed paid within a day of filing. Checking with IRS.gov, it says "taxes not processed." Why?

The Refund status page only works if you are getting a refund.

If you owe, you will see nothing. make sure your return is correct and pay the tax.

If there is an issue, IRS will contact you.

Thank you for the reply. But I didn't go to the "refund status page;" I went to my Account page at IRS.gov.

It states, "Your 2020 Tax Return Is Not Processed." 

My return was accepted electronically by the IRS; got confirmation of that back on the first of May.

you know the IRS is way backed up on processing returns, right?

I also went to my account and I see all my numbers, but no refund yet.

There is nothing you can do but wait.