The England international was forced off after just five minutes at Goodison Park following a heavy challenge from Adam Le Fondre, which left him with a nasty gash on his ankle.
"It's a bad one," Moyes told BBC Sport. "He'll have surgery tonight to stitch it up.
"I'm disappointed because Jags is such a good player. I've genuinely not seen the challenge again.
"At the time I thought it was poor. Maybe if I see it again I will think differently."
John Heitinga replaced the 30-year-old centre-back and will more than likely deputise in his absence.