Stadia is using Google Pay to do the transactions, so if your payment method (Google Play balance) is an accepted payment method in your country, theoretically it should work.
Could be that there are problems with iOS though, I mean Stadia has just started and there may be bugs, so what you could try on iOS would be to delete cache & data or reinstall relevant Google apps on your phone, just to be sure (to delete temporary/background data in these apps). Check also if you are using the most recent iOS version.
If nothing helps though I would recommend to contact Stadia support to make them aware of the issue or leave feedback for the developers. Hope this helps and you're soon able to use your Google Play balance to purchase games!
@Economybore I tried purchasing a game using Android and iOS, and doesn't look like you'll be able to use iOS to pay using Google gift cards. Do you have an Android device you can use? The option to enter gift card is available there.