The 31-year old former Leicester City, Watford and Derby County defender joined the Quakes from Coventry City earlier this month and performed a steady cameo role in his new clubs 1-0 pre-season friendly over Norwich City on Saturday.
Watson admitted the second-half showing from Stewart was exactly what his side has been looking for in recent times.
"[Stewart] has played at the highest levels, played for a long time," Watson told MLSsoccer.com.
"His experience is something that was very attractive to us - the leadership qualities, the communication on the field and just having that old head.
"He knows the game, he knows the positional stuff, the game situation stuff, when it's on to keep the ball, when it's the right decision to play it forward.
"That's just what comes forward with experience."