The short answer
The formula, for your specific case, is:
((($5,833.96 * 12) - $8,600) - $46,125) * 0.22 + $4,807.50) / 12 = $680.80
This will be true for the remainder of the year, assuming that you do not change your W-4, or have a change in income. Note that it is not exactly your withholding; I'll come back to this later.
The long answer
As @d-stanley notes, the US income tax system is a progressive system, in that you are not taxed a flat percentage on your income (specifically regarding income tax; you correctly note that FICA and Social Security are a flat percentage in general). Therefore, to estimate your income tax, and therefore your necessary withholding, the formula you're looking for is going to be a sort of step formula.
I'll walk through the formula I built above. It is derived from IRS Publication 15-T, Federal Income Tax Withholding Methods, Worksheet 1A. This is the document linked by @d-stanley; however, I believe he or she linked you to the incorrect location. The more appropriate location would be here, the section titled "1. Percentage Method Tables for Automated Payroll Systems...". (The PDF version of that publication is available here; I like the PDF versions better at times because the formatting is easier to read, and I think that's the case here. The section is on page 8).
The first part of the formula,
$5,833.96 * 12, is
income_per_pay_period * number_of_pay_periods_annually. In your case, I assumed the pay period was for the month of January; therefore, the number of pay periods is 12. This gets you your total income (from this job). The next part,
- $8,600, is related to your standard deduction, which is $12,950; the remainder ($4,350) is included in a subsequent step. The net result (
$5,833.96 * 12) - $8,600 = $61,407.52, is your annual adjusted payment amount. This is approximately your taxable income.
Given this annual adjusted payment amount, it's time to fix out the estimated tax burden. The IRS provides a lookup table for this on page 10 of the PDF, titled "2022 Percentage Method Tables for Automated...". Flipping down to the "Single or Married Filing Separately" table, we note that because the income tax is progressive, we effectively need to compute the income tax for each tax bracket. This has been done for us in Column C. Since you're in the 22% bracket (your annual adjusted payment being $61,407.52, which is greater than $46,125 and less than $93,425), the tax on your income in the 10% and 12% brackets is $4,807.50 (
($46,125 - $14,625) * 0.12 + ($14,625 + $4,350) * 0.10 = $4,807.50). Also note that the income from $0 to $4,350 is taxed at $0 (for the purposes of this table), which makes up the remainder of your standard deduction referenced in the previous paragraph.
The last part is the tax from the marginal tax bracket you're in,
(($5,833.96 * 12) - $8,600) - $46,125) * 0.22. This is your annual adjusted payment, minus the lower boundary of the tax bracket you sit in, multiplied by your marginal tax rate (the tax rate for that bracket). This tax is $3,362.15, plus the additional $4,807.50, for a total of estimated tax of $8,169.65. You can then divide this by the number of pay periods you have, 12, for a result of $680.80. While this does not equal $681, your actual withholding, the IRS permits employers to round tax to the nearest dollar; see the section on rounding (page 7 in the PDF).
Therefore, the generalized formula you're asking for looks like this:
(((pay_per_pay_period * number_of_pay_periods) - deduction_according_to_filing_status) - bottom_boundary_of_marginal_bracket) * marginal_tax_bracket_rate + tax_from_lower_brackets) / number_of_pay_periods = withholding_per_pay_period
I think this is actually fairly straightforward--take your income for the pay period, extrapolate it to a year, use information provided by the taxpayer to estimate taxable income, and spit out an estimate for the tax due for this pay period.