Can a grounding issue with newly installed light fixtures cause a short circuit/blown fuse?
Unless you connected a hot to ground, it probably is not related to the ground.  Without seeing some pictures or diagrams, my best guess is that one of your white wires is actually a hot (and someone didn't mark it as so), so by connecting the whites (which are supposed to be neutral) together, you've created a short.

Sometimes with switches, when the power comes into the ceiling box first, there will be one cable to the switch from the ceiling box. In this configuration you will find a black and white wire, but both are actually hot since the switch only interrupts the hot wire.  Take a look at the switch wiring - are both wires black, or is one black and one white? If one black and one white, then one of your white wires is actually hot and should not be connected to the rest of the white (neutral) wires.

I would start by disconnecting your work, verifying the fuse doesn't blow in this configuration and then start connecting the lights back up at one at a time.

Remember that electricity is dangerous and there's nothing wrong with calling an electrician if you are in over your head!