No, direct payment with your (credit or debit) card is currently not possible on the Île-de-France network, be it a physical card or one in your phone (Apple Pay, Google Wallet...). You need to have a pass of some sort, either a physical one, or a digital one on your phone, and load it with the right ticket(s) before your trip.
There's been talk about it for ages, but it took already long enough to be able to integrate a pass in a phone (even more so on iOS) that I wouldn't hold my breath. Paris went from being one of the very first to introduce contactless payments to being stuck in one of the most unfriendly systems you could imagine for occasional travellers and tourists (the focus was, and in the most part still is, on people with weekly/monthly/yearly passes).
There is also no equivalent to the "pay-as-you-go" option which exists in London or Hong Kong (and many others), where you preload the pass with some money, and it will deduct the right fare as you tap in/tap out. There is a vaguely similar post-pay system, Navigo Liberté+, but is not available on phones, and you need an bank account with a BIC/IBAN.
For a long time, the fare system in the region was very complex, with some fixed fares (bus, tram, metro everywhere, RER only within Paris itself) where you validated only on entry, and some variable distance-based fares (RER and trains outside Paris), where you had to validate on entry and exit. There were some trips where the cost would depend on which mode of transport you used, even between the exact same origin and destination!
Now they have simplified the system a lot (possibly too much as that resulted in a massive hike of the most common fares), and there are three possible fixed single-trip fares:
- Bus/tram ticket (2 €): allows you to make one trip using only bus and tram, including connections within 1h30, but no interruptions or roundtrips. Some exceptions apply (Roissybus, Trams T11, T12, T13, T14).
- Metro/RER/Train ticket (2.50 €): allows you to make one trip using only metro, RER, commuter trains, trams T11 to T14, including connections, within 2h, excluding access to the airports (CDG and Orly).
- Airport ticket (13 €): allows you to make a trip from/to/between CDG and Orly airports. Except via Roissybus, same price, but different ticket...
All of those can be added to a physical Navigo card, or to a digital Navigo card in your phone. You need to buy them before tapping in (you can do so directly from your phone). There are no discounts for multiple purchases (contrary to the old tickets which were cheaper when bought by 10).
Note that there are a number of incompatibilities, where you can't have two tickets of different incompatible types on your card or phone at the same time!
There are no automated caps either, and no refunds of unused tickets.
So, in general, you would just buy the right type of ticket (bus/tram, metro/RER/train, or airport) just before using it. On an iPhone it's just a few taps in the Wallet, but it's still a lot more annoying that just touching in/out with your payment card or phone and not having to think about it.
If you intend to travel a lot on one day, you can buy a "Navigo jour" (1-day pass), which will cover all travel... except airports. It costs 12 euros, and can be loaded on your phone.
Then if you really don't want to have to worry at all (but it's not necessarily good value for money -- or more accurately, it probably never is), you can buy a "Paris Visite" pass (for 1, 2, 3 or 5 days), which gives you full access to the whole network, including airports.
If you are staying for several days, weekly or even monthly passes may be a better option, but they require a bit more work to set up. A weekly pass (Monday-Sunday) covering the whole network (including airports) is 31.60 €.
Note that:
- Some types of passes require the creation of an account
- Not all passes are available via all channels. The user interface built directly into the iOS Wallet doesn't show the weekly or monthly passes for instance, you'll need to use another app (like the RATP app for instance) to buy them.
- Some passes require an account via the RATP app but not via the Wallet!
- Some channels allow you to pick the date of validity of a pass, others not
Good luck!