The resolution of your screen has nothing to do with the size of images you can create.
I can create a 1200x1600 image on a 600x800 screen. It just means I'll have to do some zooming and panning and scrolling to see it at actual size.
UPDATE:
It would help to understand how you plan on accommodating different screens. Assuming you want to make a full-screen game, there are a few scenarios you could consider:
Design your imagery at a specific size, then stretch-to-fit.
- Pros: One set of assets. One 'playing field'. If you design them fairly large to begin with, they should render fairly well even when stretched.
- Cons: You need to account for aspect ratio differences (perhaps letterbox as needed). Shrinking down too far or up too far could produce a reduction in quality that is unsatisfactory.
Design your imagery at a specific size, but then crop/expand the playing field rather than stretching.
- Pros: Assets are shown at 'actual size'. There is no aspect ratio issues.
- Cons: Depending on the design game, this may or may not work.
Design to specific screen resolutions.
- Pros: It will look 'perfect' on each screen you design for.
- Cons: It may require creating several versions of each asset.
Build the game using vector-based assets.
- Pros: Vectors can scale up and down without a loss in quality.
- Cons: Not all platforms/gaming dev platforms support them.
