There are many different rim/tyre size compatibility charts throughout the web. Most of them say you can go as low as 28 mm tyre with 19 mm inner width rim.
For clarity, I interpret the marking on the rim (622 * 19c) as meaning that the rim's internal width is 19mm. I am not sure that this is a standard marking, but the second number is likely to refer to the inner width. It could be verified if you have a pair of calipers, or even a measuring tape. If you really wanted to do this, just measure the distance between the bead hooks on the sidewall with the tire off. The "622" part is unambiguous and it means that the wheel has a bead seat diameter of 622mm (i.e. it's a 700c wheel), but you knew this already. I'm not sure what the numbers 30 and 20 mean.
The link that max gave is correct in one sense, and it is the previous compatibility standards set by the Internal Organization for Standardization (ISO) and the European Tire and Rim Technical Organization (ETRTO). These standards are old, as I discussed here, and they are likely to be conservative. That is, if your rims are indeed 19mm internal width, you could mount 23mm tires on them if, for some reason, you decided you wanted to do so. Your proposed 32mm tires are still within the official specifications, so this is just offered for completeness.
For the record, as I attested in comments on max's answer, I have a pair of Hed Jet wheels, which have 21mm internal rims. I can't find the official tire recommendations on Hed’s current site. I can attest that at one point, they were 23mm front, 25mm rear, for the purposes of maximizing aerodynamics. I can also attest that I've done a couple events with a 23mm front tire and had no trouble.
The ISO and ETRTO are in the process of finalizing tubeless rim compatibility standards. It is possible they will update the tire size compatibility standards, but I'm not certain if or when they will do so.