There is no more risk of stalling an aircraft from a TCAS alert than from being instructed verbally by ATC to climb to a higher altitude for traffic avoidance. Or, (in the case of slow flight at approach speeds) to execute a go-around from an approach to land.
An aircraft can only be stalled if the critical angle of attack is exceeded. A TCAS alert to climb is just that, it does not command the pilot to do anything other than climb to a safe altitude. It tells the pilot where to go, but not how to get there.
Initiating a climb at low speed does not automatically result in a stall as the question would seem to imply. Proper pilot technique is to add power and adjust attitude to maintain safe airspeed and AOA. The aircraft can only be stalled if the pilot pitches the nose up, fails to add sufficient power in the climb, allows airspeed to decay, and holds the attitude until a stall is reached.
So, the short answer is no.