The midfielder travelled to join up with the international side following the Cherries' 3-0 defeat to Tottenham Hotspur last Sunday.
Martin O'Neill's men are set to face Switzerland and Slovakia over the coming weekend, but, despite trying to play through the pain, Arter felt he had no choice but to withdraw from selection.
"I went over on an early flight Monday morning," the 26-year-old started.
"I saw physio and doctor when I got there and told them that I had been struggling for a few weeks with an Achilles problem, so they had a look and got me scanned.
"After that both the doctor and the boss Martin O'Neill thought the best thing to do was to use this international break to take some rest and get it right.
"I really wanted to go away with Ireland so I had tried to grit my teeth and get through it, but in the end it was just too much."
Ireland will play their penultimate set of friendlies this weekend ahead of their Euro 2016 campaign this summer.
And whilst O'Neill may well look to shuffle his pack, Arter insists he will still be given a fair chance to fight for his place.
"Of course I had the Euros in my mind," he continued.
"That's why I wanted to be fit and available to play, but the doctor seemed pretty certain that it wouldn't help.
"I said to the boss that I could play through the injury if needs be, but he was really supportive and said that I could get myself right and have another opportunity in May.
"So now I am just going to take some rest and try to settle it down. Hopefully it won't take too long."
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