The 29-year-old Northampton star is a devout Christian but he has played on Sundays before.
He faced France when Scotland opened the 2008 Six Nations on a Sunday.
Scottish Rugby Union chief executive Gordon McKie said: "We have been talking with Euan for some time now and fully respect his wishes."
Murray has suffered from several injury setbacks in recent seasons, including a broken shin bone and ankle ligament problems that cut short his British & Irish Lions involvement this year.
He returned a month ahead of schedule to help Northampton to a 15-13 win over Bath on 5 December.

However, Murray, who is reluctant to discuss his choice, will also miss any club games that fall on Sundays.
Former Scotland winger Kenny Logan is an ex-team-mate of Murray's and is also respectful of the prop's change of heart.
"It's his decision and we have to go with it," Logan told BBC Scotland.
"I played at Glasgow with Euan. He's a born-again, he's very happy, he's struggled with injuries and he wanted to change his life. He's done that.
"At Northampton he won't play on a Sunday. He's been up front with them and the club can rotate and know when to play him."
Former New Zealand flanker Michael Jones refused to play on Sundays due to his religious beliefs.
And Murray's decision is sure to draw parallels with Eric Liddell, the first Scot to win an Olympic gold medal.
Liddell, who played seven rugby internationals for Scotland, was a committed Christian and refused to run in the heats of the 100m at the Paris Olympics in 1924 because they were run on a Sunday.
He had been red hot favourite to win the event but did succeed with an unexpected 400m triumph after collecting bronze in the 200m.
Scotland begin next year's Six Nations campaign at home to France and also welcome England to Murrayfield.

"The day of 400 metres race came, and as Liddell went to the starting blocks, an American masseur slipped a piece of paper into Liddell's hand with a quotation from 1 Samuel 2:30, "Those who honour me I will honour." Liddell ran with that piece of paper in his hand. He not only won the race, but broke the existing world record with a time of 47.6

This article was originally posted on 15-Dec-2009, 20:06 by Hugh Barrow.
Last updated by Hugh Barrow on 15-Dec-2009, 20:16.