The reigning Scottish champions have announced that Lawwell has decided to retire after 17 years in the role.
McKay has been with Scottish Rugby for 13 years and was promoted to chief operating officer in 2015.
Previously he was a principal player with Pernod Ricard, who took over Allied Distillers’ the extensive, state of the art packaging plant for top drinks brands including Chivas Regal, Ballantine’s and Laphroaig Scotch whisky at Kilmalid in Dumbarton.
Lawwell said: “I believe the club have identified an excellent successor with the highest integrity.”
The 61-year-old Scot has been at the helm during a time when the Glasgow club have won 29 trophies, including 13 league titles, four domestic trebles and reached the last 16 of the Champions League on three occasions.
But Celtic have fallen 23 points behind city rivals Rangers in this season’s attempt to win a 10th consecutive domestic league title, albeit with three games in hand.
“It was obvious that the club would be facing a natural period of transition at the end of this season regardless of any outcomes,” Lawwell said in a club statement.
“Like every Celtic supporter, I am disappointed that, so far, this season has not worked out in the way we all hoped for, despite our best efforts.”
Lawwell, pictured left, who said he first discussed the possibility of retirement with chairman Ian Bankier last year, had delayed his announcement until a replacement was found and “will now look forward to other challenges”.
Lawwell recently apologised for Celtic’s controversial trip to Dubai, saying it was “clearly… a mistake” to travel to the Middle East during the Covid-19 pandemic.
Celtic’s principal shareholder, Dermot Desmond, said: “When Peter told me he was considering retirement, I did my utmost, unsuccessfully, to dissuade him.
“Celtic Football Club owes a major debt of gratitude to Peter Lawwell, who has made a unique and transformative contribution to our club.
“Over the years, Peter was sought after by many English Premier League clubs, but his love and commitment to the club and Scotland were more important to him.”
Celtic fan McKay joined Scottish Rugby as director of communications and public affairs in 2008 and, as chief operating officer, has led the governing body’s commercial, marketing, ticketing and communications activity and management of Murrayfield Stadium.
He has also overseen the commercial activities of Scotland’s two professional teams, Glasgow Warriors and Edinburgh Rugby, and is also current chair of the Guinness Pro14 league.
McKay said in a statement: “Joining Celtic FC as chief executive is a once in a lifetime opportunity. I have been incredibly impressed with the vision and people within the club and I look forward to working with them.”
Celtic Park before a big match. Former Dumbarton-based Dominic McKay has been appointed as the club’s chief executive. Pictures by Bill Heaney
Scottish Rugby chief executive Mark Dodson agreed that it was a “fantastic opportunity” for McKay, who he said had “made a significant contribution to the progress we have made as a sport and organisation, including recently helping to bring significant private equity investment into the Pro14”.
“Scottish Rugby is currently entering into a period of strategic review that will see the organisation plan for its post Covid-19 future and I will reference this work to reflect the needs of the organisation ahead of any recruitment process,” he added.
The Pernod Ricard plant, brands and offices at Kilmalid in Dumbarton.
‘A troubled few months’
BBC Scotland sports news correspondent Chris McLaughlin
Peter Lawwell’s time at Celtic Park was undoubtedly a successful one. Since his appointment in 2003, the club has secured a total of 29 trophies, including 13 league titles and four domestic trebles.
He has also been highly influential in Scottish football, with various spells on the boards of the Scottish FA and Scottish Professional Football League.
But the club has had a troubled few months. An ill-fated winter training trip to Dubai drew criticism from First Minister Nicola Sturgeon and fans have called for the sacking of manager Neil Lennon after falling 23 points behind rivals Rangers.