Your sentence is correct as is, but I think "suggest" can be used with an object. It's not inappropriate to say:
"I can see you're not well and so I suggest a visit to the doctor." "I had a cold and my mum suggested chicken soup." "Of all the cars you viewed, I suggest the blue one."
User_garyCan I say, "I suggest you to visit the doctor"?
No.
Your original sentence had a tacit "that" in it. It is the same as "I can see you're not well, and so I suggest that you see a doctor." You can't say "I suggest that you to see a doctor."