Interesting idea! I’ve noted the request. 
While this isn’t currently possible with the traditional Weather Tile, you can do it with any Media Tile since the Media Tiles have support for opening a hyperlink as an action. And since Meda Tiles are commonly used for displaying weather forecasts, this just might work in your situation. 
Here’s a neat example where upon clicking on a forecast (Media Tile) it opens Windy.com with a neat weather animation within a modal. The same concept could be used to open an app in a new window too!