The writer of the hit crime show, which starred Idris Elba and Dominic West, was commenting on Morgan's appearance with Fox Nation presenter Tucker Carlson to discuss why he left Good Morning Britain over Meghan Markle's interview with Oprah Winfrey.
Simon tweeted: "In an ocean of talking-head sewage, what are the chances that the two most pouty and puckered a**holes in broadcast television would ever swim through the brown, find each other, and discuss the events of the day?."
Morgan, 56, hit back: "Talking of sewage, Dave... you were all over me like a sycophantic chap rash when I interviewed you. Yet now you play the vicious horrible illiberal liberal to appease your vile woke base. All a bit s**t really. But on a positive note, I loved The Wire."
The pair went on to exchange insults, with 61-year-old Washington-born writer Simon dismissing their meeting as simply a press question-and-answer session at which he was being polite.
Watch: Piers Morgan says members of the Royal Family have thanked him
Simon retorted: "You met me once, on a stage at Cannes with Richard Plepler & Aaron Sorkin on a panel I attended at the request of HBO. 1) Didn’t know you from a fart in high wind. 2) Was polite to said fart at Richard’s request as that man was a Medici to me. Now go lick some racist royalty."
TV and film director Aaron Sorkin created US political drama The West Wing. Richard Plepler is the former chairman of HBO.
Tabloid editor turned TV presenter Morgan responded: "Aaron was a class act. You.... not so much. Probably explains why The West Wing is so much better than The Wire."
Simon then referred to Morgan's failed US chatshow Piers Morgan Live, which was axed in 2014 due to poor ratings.
He tweeted: "Sorkin is swell and has gigs. I’m still working for Time Warner after a quarter century. You’re unemployed and if it was a Tijuana cathouse and you had a fistful of fifties, you still couldn’t get laid at TW or anywhere else over here. Not after that pratfall you did at CNN.
Morgan's response was to highlight his 7.9m Twitter followers, while Simon has 313.4k.
The former GMB host said: "Yet oddly, despite your stupendous ego, I have 26 times as many followers as you! You’re a great writer but a repulsive human being. Aaron’s a better writer and a great guy. Be more like Aaron."
Simon replied: "Yes, popularity on this platform is truly indicative of worth and mettle. It’s a damn wonder you keep getting fired when actual journalism is required.
"Now go back [sic] some poor girl’s cellphone and call that moral midgetry reporting, you empty parasite."
Morgan has admitted that as editor of the Daily Mirror he listened to messages on the phone of Heather Mills, Paul McCartney's ex-wife, when it had been hacked by Mirror journalists.
His former paper the News Of The World hacked murder victim Amanda Dowler's phone after she disappeared in 2002 but Morgan was no longer working there at the time.
Simon rose to fame following the success of The Wire, which ran for five seasons on HBO between 2002 and 2008.
His most recent series The Plot Against America aired on HBO last year.
Watch: Piers Morgan's interview with Fox Nation