At some point however it seems like they reach their maximum size and they stop scaling.
Thank you for providing the screenshot.
In the screenshot, there really is not enough room to display a 4th thumbnail on the 1st row (so Showkase wraps the 4th thumbnail onto the 2nd row).
Here's a mock-up (click here for screenshot [Link removed.]) of how the page would look with a 4th thumbnail in place.
The thumbnails are responsive but they will reach a maximum size and will not continue to scale up to exactly fill the available page width. (They will not be scaled up beyond their actual size as this would degrade their visual quality.)
I think it would be very difficult to change the way the the gallery index page thumbnails are scaled in the Showkase source code. I expect you'd need to change several files, knowledge of PHP and CSS would likely be necessary and you may encounter problems (such as with breakpoints) along the way.
You are certainly welcome to try but I'd recommend sticking with the way that Showkase currently works.
Having said that, you could try changing the dimensions for the thumbnails (to compensate for your increased page width) via the ss_totalIndexThumbWidth value in the '_themes\base\theme.ini' file (on line 27). (Please note that the line number refers to the current version of Showkase, v1.7.2.)
This might help to space out your thumbnails a little better to give a more balanced looking page layout.
Be sure to republish your site after making any changes.
When I do the same however on my phone I first get some "in between" screen with the first photo of the gallery, a text with the number of images and "view gallery".
What you are describing is the Splash Page.
The Splash Page is a placeholder for the gallery which is displayed by default on small screen devices when the gallery is embedded in a web page alongside other content (rather than displayed on a page of its own with dimensions of 100% x 100%, filling the browser window) and may may too small to be usable.
The Splash Page is essentially an image link for the gallery which displays an image (by default, the first image in the gallery) and some text.
When the user clicks or taps the Splash Page, the gallery is expanded to fill the browser window (giving the images more space to be displayed).
More information about the Splash Page can be found in the Screen Modes section of the Gallery Tour.
You can choose to not use the Splash Page by setting showSplashPage="NEVER" ('Splash Screen (Pro)' section). Using this setting, the Small Screen Mode thumbnail page would initially be displayed instead of the Splash Page on small screen devices.
Alternatively, you could force the gallery to be displayed in Large Screen Mode (which, by default, does not use the Splash Page) on all devices and in all browsers by setting screenMode="LARGE" ('General (Pro)' section).
For reference, all the Splash Page configuration options can be found here.