Can I deduct the gross receipts tax I pay as a business expense?

There are two ways to answer this question.

State law

The first is based on state law that says gross receipts tax is the responsibility of the seller. You do not act as a conduit that passes through the tax from the buyer to the state. (That's different from most other states' sales tax laws.) This requires that the sales amount on your federal income tax return is the same as the total you receive from your customers whether or not you added tax to the sale; and you deduct the tax at the time it is paid. (This follows cash basis accounting principles.)

Sales on federal tax return equals reported gross receipts

The second way is that the sales you enter on your federal return is the same amount you report on your gross receipts reports filed with the state. It's the amount after the tax has been removed arithmetically. If you do it that way, then the tax is not a deduction.

Which is correct?

The first approach is technically correct because it follows state law, while the second one is more practical because the sales on on you federal return will match the gross receipts reported to the state and thereby avoid a discrepancy that would cause the state to want to audit.

Either way you will end up with the same taxable income on your income tax returns.