Evans has played 28 Davis Cup games for Great Britain and was a substitute when Great Britain won their last title in 2015.
However, he was brushed aside in the first set by Shapovalov, who silenced the 15,000 spectators at Manchester Arena by scoring the first set in just 25 minutes.
The 34-year-old Brit finally got on the scoreboard when he held serve at the start of the second set but failed to make the most of two break points at 4-4.
Former world number 10 Shapovalov backed up a dominant first set as he scored the decisive blow 6-5 to take his country into the knockout rounds, which will take place from November 19-24 .
“I thought long and hard about it. I need to think a bit more,” Evans told BBC Sport.
“I love competing for my country. I loved every minute of it, but sometimes you have to know when to go.”
“It will be a difficult decision, but sometimes it’s nice to give someone else the opportunity and watch from afar.”
Evans has fallen to No. 178 in the world rankings after deciding to miss some tournaments this year to concentrate on playing doubles with Andy Murray at the Paris Olympics.
“His ranking has gone down so he has a lot of things to consider – what’s best for him now and what does he need to have a successful year next year,” Smith told BBC Sport.
“Whatever happens, he has been a great servant to us.”
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