Apple pay is a combination of hardware for authentication and security, with a unified payment interface on the backend. It makes sense as a mobile payment solution.

Using your device to authenticate web purchases is a more complicated and difficult (and potentially insecure) prospect, and thus, will take longer before it's viable.

Apple's normal mode of operation is to start with the smallest essential thing that adds value, and then extend it over time. In a way, what we saw today is Apple's idea of an MVP.

If the web isn't an area where Apple can add value, they won't do it. Or put another way, if other people do a better job on the web (e.g.: stripe) then Apple's not going to try and compete with them.

Web is the enemy of Apple. iOs devices need unique features to sell, not Web stuff that is easy to duplicate by the competition.