Fuse for motor protection?

No internal overload protection? It won't hurt anything to fuse it but you may be adjusting the fuse size.

As in a resettable overload button? Don't remember, I'll check. Can you give me a couple of "what if's"? ("If it does, then..." and "If it doesn't, then...")
As in, the 9A may blow quickly and I'll have to adjust upward a little?

Any reason that you didn't bother to disclose all the important motor characteristics and details? Single phase or 3 phase? 1725 rpm or something else? The nameplate of your 1/2 hp motor should specify the operating current at 208v. What is the nameplate operating current? Extra credit for the maker and model number.
The problem is that starting current can be 3 to 4 times the nameplate oprating current. Your common 9A slow blow cartridge fuse isn't going to work if your motor operates at about 4A.
You have some choices. Try plugging into this calculator and pick the fuse type (non-time delay, dual element time delay, instant trip, or inverse time trip) that you find appropriate.

And does the motor have to start-up fully loaded

If it has internal overload protection you are only protecting for a short circuit and the 20a branch circuit breaker is usually sufficient. (less than 2.5x the motor FLA)
Yes the 9a fuse may not be enough to hold the starting current over time or it might be too much to protect from overloads. That is why motor manufacturers will usually install internal protection on these small motors. Then the overcurrent device only has to protect the wiring up to the motor and protect against an internal short in the motor.

No go right ahead :-) However I reckon you would be better off with thermal protection if the motor does not have it.

How does thermal protection work? I understand the safety mechanism, but if a motor is heating up under load, isn't it drawing over-max current? Shouldn't the (carefully-chosen value) fuse blow?
I need to learn about this, I know...

The thermal is actually looking at the winding temperature and a marginal overload that accumulates heat in the winding over time will cook off the thermal where a fuse might not go.
It is fairly unusual for fractional HP motors not to have a thermal. They are generally designed for connection to regular 15 or 20 amp circuits. It should be indicated on the motor labels.

The phase angle of induction motors changes with varying load. The current draw only changes a little with loads from zero to the motor's rating. So, if the motor is rated at 1/2 Hp, 208 V, the running current at rated load will be something just over 2 A. I = (745 W * 1/2 * 1.2 (efficiency))/208
At idle, the current may be 1.5 - 1.7 A with a very low power factor (phase angle near 90 degrees lagging). At full load, the current will be about 2.1 A with a phase angle of maybe 5 - 10 degrees. This, the motor draws more REAL POWER at full load, but the current doesn't change much throughout the range the motor is rated for. This is why external fusing for motors is to protect the building wiring from fire, it does not protect the motor from minor overload or overheating. Internal thermal protectors actually measure the temperature INSIDE the motor, and thus are a lot more effective.
Jon

There's usually a fair bit of heaving and grunting as a motor starts spinning - especially if it starts under load.
Even a slow-blow fuse that can handle the second or two of what is pretty much stall-current, will carry enough steady running current to let the magic smoke out if a fault/overload develops.

A fuse is not a replacement for a properly-sized motor OLP overload protection device.
A fuse will protect wiring, as previously stated.. but you'll just keep asking the question until you get the answer you want, right?
Overload protectors and TP thermal protection devices aren't the same, and are intended for completely different situations.
The OLP will react rapidly to protect a motor in the event of a sudden machine jam or other fault.
The application was previously stated to be a bandsaw which previously had a smaller motor on it.
Generally, fuses aren't motor protection devices, and won't replace a proper OLP.