Not by itself. A statement proves you paid a merchant, not that the payment was for qualified medical care.
A card or bank statement shows a merchant name, a date, and a dollar amount. It does not show what you bought.
"CVS/PHARMACY #4821 — $63.40" is consistent with a prescription. It's equally consistent with shampoo, a birthday card, and a bag of chips. That ambiguity is the whole problem: qualified medical expenses have to be qualified medical expenses, and the statement can't establish that.
Statements are still useful as a supporting document — they corroborate the date and amount on an itemized receipt, and they're an excellent tool for finding expenses you forgot about. Pair them with the real thing.
If the itemized receipt is gone, most providers and pharmacies will reprint one. Pharmacies in particular can produce a full annual prescription history in a few minutes at the counter or from their app.