Assuming you control the web page being shown on the public display, yes.
The web page would need to either periodically contact the server via AJAX, or have a long-running connection to the server (i.e. Comet or WebSockets).
When the server receives the request from the mobile device, it either uses the Comet connection to send the new URL to the web page, or when the web page next contacts it via AJAX, it sends the new URL in response.
The web page then sets its own window.location
property to this new URL.
Note that once it's done this, you'll no longer be able to send the browser in question to another new page, unless the page that you've just sent it to also includes the JavaScript that contacts your server.
But if you don't control the web page being displayed...
Then you'd need a browser extension to initiate the connection between the browser and your server.