Yes you can use Apple Pay with Craft Commerce and the first-party Stripe Gateway Plugin.
In order to do that, you need to customise your template and add the functionality for the Payment Request button. What this does is more or less generating a charge promise token for Stripe (same way it does for card).
Charging the token is then handled by the gateway itself as you would with a card charge.
You can choose to display the button by checking if a relevant Payment API is supported with paymentRequest.canMakePayment() method.
I hope this helps in terms of a high overview.
The above answer was for use with Craft Commerce 2, as originally set by the question.
Requirement changed to Commerce 1.
The quick answer is still yes.
We have completed a similar task before (with Stripe Elements and Stripe Sources), though that's before SCA.
But Elements are SCA ready as far as I know and you need to use the PaymentIntents API PaymentMethods API.
Difference is that the main Stripe Gateway for commerce 1 (by omnipay) doesn't seem to support webhooks from Stripe, which informs your system whether a token (Source at the time) can be charged or not.
Hence, you need to build kind of your own plugin that loads an extra stripe gateway either new or extending the existing Stripe gateway class (extending is easier as you don't need to rebuild everything).
Might need to add a custom Payment Form to your plugin.
- You still need to build a custom template.
- The template should use Stripe elements.
- Logic is same, chargeable promise token which you need to charge.
- Request for charging
- Await webhook that notifies your system if your intent can be charged
- And then continue with charging if available, otherwise inform user for any errors.
That's still a high overview of how the process works.
You would need to use Payment Intents API and Payment Methods API