The 28-year-old – who has signed a pre-contract agreement to join Juventus at the end of the season – has withdrawn from the squad and returned to his club for treatment.
Ramsey played no part in The Dragons' 1-0 friendly win over Trinidad and Tobago on Wednesday.
Wales manager Ryan Giggs said: "It's a blow, especially the form he's been in, but it gives a chance for somebody else.
"We made sure we had a plan B and we've worked on that during the week. We're going to miss someone like Aaron, but we've prepared for it as well."
Ramsey will hope to return to full fitness for the Gunners' next game against Newcastle United in the Premier League on April 1.
The Gunners midfielder was among a host of big names rested for the Trinidad and Tobago match, with Gareth Bale, Ben Davies and Joe Allen all amongst those rested by Giggs.
Despite toiling to victory, the manager will be confident he made the right choice by leaving Ramsey in particular out for that match-up as he and his charges prepare for Sunday.
"We've got players minutes who needed it, a lot of lads who weren't even on the bench are fresh," Giggs said after the clash.
"Slovakia are playing on Thursday, and it's not a given but we wanted to use that advantage. Going into the game fresh, with a win, there's a lot to be pleased about.
"I thought there was some really good stuff. We've only had a few days to work on stuff and some of it paid off.
"Second half I think the lads tired, which you can understand, and credit to Trinidad and Tobago they just sat in their shape and waited to counter-attack.
"But we kept going and got our reward in the end. We had our luck a little bit with Gunter clearing it off the line but I told the lads if we kept a clean sheet we'd always have a chance."