Ledley, 29, broke his leg in early May but was named in the Wales squad and came off the bench against Slovakia.
"Without Sean, I don't think I'd be here," said the Crystal Palace player.
"I'm in one of the best shapes of my life now. I've worked so hard. I've been in an oxygen chamber, up at six o'clock in the morning and not getting home until four or five for weeks."
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Wales physiotherapist Connolly is also part of Crystal Palace's medical team and has helped Ledley's rehabilitation since he suffered the injury against Stoke City on 7 May.
The former Cardiff City player came on in the second half against Slovakia and played a part in Hal Robson-Kanu's winning goal.
Ledley's Crystal Palace team-mate and room-mate Wayne Hennessey missed the game in Bordeaux after suffering a back spasm and was replaced by Liverpool goalkeeper Danny Ward.
Hennessey missed training on Monday but Ledley is hopeful the 29-year-old will recover in time to face England in Lens on Thursday.
"Everyone was gutted for him because he deserves to be playing," Ledley said.
"But these last few hours he's moving about and hopefully in the next 24 hours he can get on the training pitch.
"He's worked so hard and been in fantastic form for Palace."
Wales midfielder Joe Ledley says an oxygen chamber, dawn rises and physio Sean Connolly got him to Euro 2016.