You could try vinyl plugs that can be found in hardware/home improvement stores. The plugs come in various diameters. If there isn't one for the hole's size, you may have to drill it larger to the size of the plug so that it will fit.
Also found this kit, may be this will be an alternative option.
Obviously the best bet would be to replace the tile, if it's tile, if it's sheet vinyl well...
If you have ANY of the material left over, you can cut out a patch and replace it, use a liquid seam sealer to make it look good...
You can purchase a vinyl sealer, that will basically attempt to fill in the gouge, use of lacquer thinner is required, as this is going to clean the area first...
You can purchase wood putty, or a wood crayon at your local hardware store. I used to work in Lowe's flooring department. They also sell repair kits, but you would probably be better off matching it yourself, start low.
So clean it up, broom and mop. Neutral cleaner if necessary,
Apply the putty or crayon or whatever you will use to fill the gouge. They sell markers that help you fill in the extra stuff colorwise...You can set and let dry this way, wipe it clean and repeat the process, if necessary...OR...
you can apply wax to the whole floor, this will seal up and if you go low-gloss, you might be able to mask up the imperfections in a similar process as would be handled by a flat paint on an ugly wall.
Minwax has a reputable putty here
Best Answer
I mentioned a longer screw in my comment, I need to expand on this. A longer screw of the same size may need to be inches longer to contact wood framing because of shims. But a longer screw that engages with framing will not affect the gear box functions. A larger screw as some suggest may affect the gear box. I have used longer screws on high precision gear boxes that worked as well or better than the original that stripped out. But going larger may cause offset or problems with the gears in simple from the larger head and or shaft.