Will a 12 AWG circuit handle a TIG welder rated at 28 amps when set for 220V?
Close, but not quite

Given the 60% max duty cycle figure you stated for your welder, and a rated primary current of 28A, your circuit ampacity comes out to 28 * 0.78 = 21.84A.  In other words, to run your welder at that 60% duty cycle, you need 10AWG wire with a 25 or 30A breaker; or, you can reduce the duty cycle to avoid overloading the circuit, as going down to 50% duty cycle gets you to just under 20A given the 28A rated primary current of your welder.

(P.S. if you're wondering where in the Code this lives, it's 630.11(A) and the associated Table.)