The Arsenal defender had to be substituted during Wednesday's 1-1 draw with Switzerland after over-stretching for a ball, and it appeared that he had damaged his hamstring during the incident.
Tierney has been ruled out of Scotland's next match against Hungary on June 23, and the 27-year-old will now undergo a series of tests to determine the extent of the damage.
"Yeah, definitely out. It looks pretty bad. Obviously, we have to assess it but Kieran won't play the last group game," Clarke told reporters after the clash with Switzerland.
"You have to feel for Kieran. He'd got himself in great shape. A top player for us, he's doing ever so well. It's a shame but someone else will have to step in and step up to the mark."
Scotland suffer Tierney blow in Switzerland draw
Scott McTominay sent Scotland ahead in the 13th minute of Wednesday's clash with Switzerland, but Xherdan Shaqiri levelled the scores in the 26th minute.
Scotland are third on Group A on one point, but a win over Hungary in their final group fixture could still see them progress to the knockout round of the competition.
As well as Tierney, Clarke's side will be missing Ryan Porteous against Hungary, with the defender handed a two-match ban following his red card against Germany in the opening match of Euro 2024.
"He's not been sent off in a couple of years. He has been great for us since I gave him his chance against Ukraine. The challenge in Cyprus you mention (which prevented a goal), was clean, he didn't catch an opponent," said Clarke when asked about Porteous.
Tierney, Porteous will be absent against Hungary
"The other night, he was trying to stop a clear goal-scoring opportunity and he definitely went in too hard and we are pleased that Gundogan appears to have no lasting damage. It is something that Ryan will learn, but is an honest challenge to try to get the ball.
"To try to stop a goal-scoring opportunity so I wouldn't be too hard on him. He was one of my cuddles. I explained the situation. It doesn't look good.
"Ryan didn't want to make that challenge, he wanted to stop a goal. He is very down as you would expect, but we will pick him up, don't worry about that."
Tierney spent the 2023-24 campaign on loan at Real Sociedad from Arsenal, making 26 appearances in all competitions, but he is due to return to the Gunners for the start of pre-season training.