The tweets came in response to a picture that Ramsey had posted of himself attending a golf event.
"I don't wanna be rude mate but you look like a rapist on that picture! Lol," Szczesny originally commented.
"Looking like a rapist/pedofile (sic) isn't a good thing either."
The 21-year-old goalkeeper was reminded by the club of his responsibilities when using Twitter and apologised for his remarks.
"I'm sorry if anyone was offended by my previous tweets," he posted later.
"There are some things we should not joke about and I have crossed the line. Sorry!"
Szczesny has since deleted the offending tweets.