I now take back my earlier suggestion that the .177 calibre probably started in Germany around 1880-90, as 4.5 mm. I had forgotten that examples of the American Haviland and Gunn air pistol (or perhaps more accurately referred to as the Morse pistol) from the 1870's are known with .177 calibre.
So could Haviland and Gunn not only have been the first to introduce break-barrel airguns, but also the first to popularise the .22 and .177 calibres?

In the 1870's there were really only two companies driving the popular airgun manufacturing scene, Haviland & Gunn favouring 0.177 and 0.22 calibres, and Quackenbush (who was heavily into airgun pellet and dart manufacture) favouring 0.21. When the break-barel design migrated to Germany and the Gems started to appear in Europe, it seems that the H & G calibres were preferred and won the day.

I can understand that gunsmiths experimented with 0.20, 0.21, 0.22, 0.25 calibres, and probably also 0.23 and 0.24 at some point, and by chance 0.22 became the most accepted, but I can't understand where .177 came from. Why not 0.18 inches, which would be the next size down in this series?

The only logical explanation I can think of is that the calibre did start off as a perfectly logical 0.18 inches (probably by H & G), and when their designs migrated to Germany this was metricised to a nice round 4.5 mm (it would actually have come to 4.57mm without rounding down). Then after the German Gems has been imported into the UK for a few years and British gunmakers began to make ammunition and even their own guns to this calibre, they converted 4.5mm back to inches, which would be 0.177 inches if done precisely.

It would be nice to have old catalogue information about the H & G Morse pistol which specifies the calibre, but I do not know of any such thing. If it did, the chances are it would say 0.18 rather than 0.177. So according to my theory, the 0.177 calibre arose by a sequence of misleading events ( a bit like a Chinese whisper).