James Watt didn't finish his apprenticesip in London. He was not free of his indenture so he couldn't work in the trade in that town.
A friend of his got him a job as an instrument maker at the university at Edinburg. It was in the course of repairing a demonstartion model of a Newcomen engine that Watt assembled the ideas and knowledge of the heat losses of this engine that led him to develop the Steam Engine and subsequently patent the device.
Jmaes Watt is credited with the invention of the Steam Engine Indicator that plots a graph of steam pressure in the cylinder versus piston position.
This graph is a true analog of the conditions inside the cylinder at any point in the piston stroke.
The comparable electrical instrument is the ocilloscope. You can see the degree of sophistication of the man's mind.
Watt is also given the credit for the invention o the fly ball governor. He could have very easily have figured out the gvernor on his own.
There is some concern that James Watt might have seen a fly ball governing device elsewhere used to maintain the speed of a water wheel.
Historical documentation is unclear, so Mister Watt isn't given clear credit on the governor although it bears his name.
Considering his background and his mental ability, James Watt could have very well invented and made a micrometer.
Personally though, I have two questions:
1. Watt's engies were big and crudely made. If Watt was going to make an efort toward precise measurement, I would think his instrument would be one that could measure longer lengths and larger diameters.
2. Watch and clock making were already a high art in London in the middle and especially the last quarter of the 1600's (see Thomas Tompion). Although the work was small and fine, the tools were simple and somewhat crude. One would think that a micrometer device would first appear at a watch maker's shop, not a steam engine builder's.
For those interested, "The Dictionary of English Biography" gives a rather finely detailed history of James Watt's life. Many of the comments were written by Watt's contemproaries. In all my research on the subject I find this to be the best of all of his biographies.
The Dictionary of English Biography is in the reference section of most large librbries. It is a multi-volume set.