I am just trying to determine if any safety needs to be built into the circuit. I was going to size the cables properly. I thought the fuse could provide an overcurrent protection. is this not the case?
You size the fuse to protect the cabling, and make sure the cabling is generous for the load. Fuses are not meant to be precision devices, and a 1A fuse can be perfectly fine with 2A for a surprisingly long time, and conversely, blow at 0.9A if you keep it running for long enough. Thus, if your load is 1A, you'd stick in something like a 2A fuse, and cabling rated to more than that.
If you want to protect your device from a fault condition that
isn't massively excessive current draw (5A for a nominally 1A load, for instance), you need something else. If your worry is overheating: Get a thermal fuse, or monitor the temperature somehow. If you're specifically worried about the current: Get an electronic fuse; it'll have a sharp cutoff point.
This is a classic case of "I am doing something and X sounds like a good solution, and people say nice things about X so I'm going to do that. You cannot seriously be saying that X isn't going to do the thing I want it to do?". Tell us what you're doing instead of (or in addition to) how you want to solve it. In this case: What's the load, and what's the safety concerns with it?