Instead of 'calling' a page, you should execute the action method in the Controller class that is responsible for that page. If you look at the page you mentioned, at the top you should see something like <apex:page controller="myController" ...
where myController
is the name of the class that contains logic for that page. Unless the button you were talking about is a standard action (like Save or Delete), there is a custom action defined in that Controller class and it is linked to the button you mentioned via action="..."
attribute on the <apex:commandButton>
tag.
Now that you know the Controller action that is responsible for handling a button click, you just have to create an instance of that controller in your schedulable class, initialise it with Case ID and call that action programatically.
P.S. to make the solution more future-proof and avoid hitting governor limits, instead of reusing existing action and potentially executing DML statement for each individual case that you loop through, consider modifying replicating the logic of action function in your scheduled class but modifying it so that it can handle a list of cases as an input.