Does one open a browser “on” a URL?
Solution 1:
Preposition on may be acceptable in that context, but is less suitable than with, using, or at. You might also reword the sentence, for more-proper semantics:
It opens the browser to display the page at URL ...
Solution 2:
Yes, you may. With or at are more common, but a Google search for "browser on url" shows plenty of hits with this usage, for example: How to open a browser on a given URL