The following may add a bit more into the story:

The earliest airgun catalogue I have is for Hawley's Patent Target air pistol, otherwise known as the Kalamazoo, from 1870, and the pistol was advertised as available in three caibres .22, .26 and .28. [The calibres are actually expressed as 22-100, 26-100 and 28-100, in other words 22 hundredths of an inch etc.]
A later catalogue, from 1880, advertises the Haviland & Gunn air rifles (which pre-date the Gems) in calibre 22/100. Quackenbush on the other hand at that time sold his airguns in "21/100" calibre. The .177 calibre is not mentioned anywhere.

I am not suggesting that Hawley originated the .22 calibre, but it does seem that there were a lot of arbitrary calibres around by 1870, so could the .22 size have become dominant just by chance?

The first catalogue references I could find for .177 (as the metric equivalent 4.5mm) were actually German, and this calibre was evidently in common use in Germany for airguns by 1890. Possibly German manufacturers introduced the calibre around the time that mass produced airguns first came onto the market in the late 1870's, when the Germans adopted the Haviland & Gunn designs. If they wanted something substantially smaller then the American .22, then 4.5mm would have been a reasonable size to choose and is a more or less round number when expressed in mm. With the subsequent importation of Gems from Germany into the UK, the 4.5mm would then have become better known as .177 inches.