Well, it's not illegal. No crime was committed.
It might be a civil tort, but realistically, you're talking about, at most, a few hundred dollars. You will be in small claims court, and the burden of proof is on you. IANAL, but to me, your chances of prevailing are pretty slim.
You will effectively have burned all bridges, and you will be remembered as "that guy" and no one you know there will ever provide a positive reference for you.
Employment is a business transaction, not a reward system. Employers pay what they must, not what they should. If the supply / demand ratio changes, adjustments are made, and that's what a raise is for - either your service is more in demand (as your skills and efficiency improve), or market forces have made demand for your existing skill set higher. A "raise" is to keep you as an employee with the changed business climate.
You have declined that offer. The raise was not sufficient, as you took another job, so why would the employer expect to pay more when you have not met your side of the bargain?
That "Paperwork" is probably just a collection of HR forms for your supervisor to approve and you to acknowledge the new pay rate. Trying to construe that as a contract, especially when you are no longer wishing to remain as an employee, is a longshot at best.
Hire a lawyer and sue if you wish. You have that right. You're trying to ice skate uphill, though, IMO.