The word micrometer is a neoclassical coinage from Greek micros, "small", and metron, "measure". Merriam-Webster Collegiate says that English got it from French and that its first known appearance in English writing was in 1670. Neither the metre nor the micrometre nor the micrometer (device) as we know them today existed at that time. However, humans of that time did have much need for, and interest in, the ability to measure small things, and small differences; the word no doubt was coined in reference to this endeavor, even if it did not refer specifically to its present-day senses.
The first ever micrometric screw was invented by William Gascoigne in the 17th century, as an enhancement of the vernier; it was used in a telescope to measure angular distances between stars. Its adaptation for the precise measurement of handheld objects was made by Jean Laurent Palmer of Paris in 1848; the device is therefore often called palmer in French, and tornillo de Palmer ("Palmer screw") in Spanish. (Those languages also use the micrometer cognates: micromètre, micrómetro.) The micrometer caliper was introduced to the mass market in anglophone countries by Brown & Sharpe in 1867, allowing the penetration of the instrument's use into the average machine shop. Brown & Sharpe were inspired by several earlier devices, one of them being Palmer's design. In 1888 Edward Williams Morley added to the precision of micrometric measurements and proved their accuracy in a complex series of experiments.