The 36-year-old world number 73 was in line for an automatic place after McIlroy withdrew on Wednesday because of fears about the Zika virus.
But he said he would not compete because his wife is due to give birth.
"I made the decision many months ago, before I was on the team, that I would not travel," he said.
In addition to McIlroy, Fiji's Vijay Singh and Australia's Marc Leishman have already pulled out of the Games because of the Zika issue, while major winners Adam Scott and Louis Oosthuizen have chosen not to appear for scheduling and family reasons.
World number one Jason Day says he will talk to his family about the risks of the Zika virus before deciding whether to play at the Games.
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US-based McDowell, who won the US Open in 2010, says his decision centred around his wife Kristin being due to give birth to their second child shortly after the Olympic tournament from 11-14 August.
His statement added: "I have always been a proud member of Ireland's golf teams, from my amateur career through to playing in four World Cups of Golf and I wish the Irish Olympic team the very best of luck in Rio."
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Former US Open champion Graeme McDowell has ruled out the possibility of replacing Rory McIlroy for Ireland at the Rio Olympics this summer.