McIlroy birdied four of his last six holes at the US Open at Erin Hills to improve seven shots on his opening 78, but still missed the cut by four shots.
That appeared to prompt Elkington to write on Twitter that ''Rory is so bored playing golf...without Tiger [Woods] the threshold is prolly 4 majors with 100mill in bank.''
Four-time major winner McIlroy swiftly responded with: "More like 200mil... not bad for a ''bored'' 28 year old... plenty more where that came from'' and posted a screenshot of his Wikipedia entry which lists some of his achievements in the game.
But speaking ahead of his title defence in the Dubai Duty Free Irish Open, which benefits his own foundation, McIlroy admitted he regretted responding to Elkington and was avoiding social media for the foreseeable future.
"I must have wrote that tweet and deleted it about five times before I actually sent it," McIlroy said at Portstewart Golf Club.
"I sort of regret sending it at the end but I actually gave my wife, Erica, my phone and my Twitter and told her, 'Change my password to something else and don't tell me what it is'.
"So as of the time being, I'm off social media just because of that reason. I don't need to read it. It's stuff that shouldn't get to you and sometimes it does.
"It's not what was said. It was who said it and anyone that's been in that environment should realise how hard golf is at times. That's the thing that got to me more than anything else.
"If it was written by a member of the media or something I could let it slide, because I can sort of say to myself, they don't really know how it is and they don't know what you have to deal with.
"But a former player that has won a major and been successful, that's sort of why it got to me and that's why I sort of retaliated a little bit."
McIlroy has recorded five top-10 finishes in majors since his last victory in the 2014 US PGA without seriously contending, but is optimistic about adding to his tally despite an injury-hit campaign.
"I'm close," the world number four added. "I've got two big opportunities coming up this summer. Birkdale is a golf course I feel I can do well at.
"And then Quail Hollow for the PGA, I've won there two times, my first win on the PGA Tour and last time I won by seven.
"I think they are two opportunities for me to get back on the major ladder and start to win those again."
After missing the cut in the Irish Open for three years running, McIlroy won the title at the K Club 12 months ago thanks to a birdie on the 16th and a spectacular eagle on the last.
And he believes he has learnt how to handle all the off-course distractions which come with being the tournament host, which included a Q&A with Manchester City manager Pep Guardiola in Belfast on Tuesday evening.
"I'm very proud to be sitting here as defending champion," McIlroy added. "The Irish Open is a title I've wanted to win since I was a young boy, and to realise that dream last year was very special. Not only because of just winning, but in the fashion that I won, and it was definitely one of the highlights of my golfing career.
"There's always that little bit of extra pressure when you come home to play and I've struggled with that in the past. I felt last year was one of the first times I embraced it and used the crowd support in the right way rather than it being a negative.
"I've played quite a bit of links golf over the past 10 days since being home. You never quite know till you're out there in the heat of competition and I don't want to talk myself up too much, but my game is feeling good."
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