I accepted a new job offer. Now my current employer has offered a promotion and more money. How to gracefully back out of the new job?

Hopefully you haven't signed an employment contract that lists specific penalties with this new company.

Just tell them thanks but no thanks. That is, if you haven't signed anything.

If you have more details are needed.

But if it is just an open ended employment contract, giving 1 week notification will work. Again, that is assuming your contract doesn't state how long you must give (here in Ontario its 1 week if you have been working with the company less than 5 years I think)