Is the use of "on" here correct?

OED gives the following relevant sense:

9. On the occasion of (an action); immediately after (and because of or in reaction to); as a result of; = upon prep. 7.

That fits your sentence just fine. The grammatical problem arises rather with the next two words. “On meeting” is a front-shifted modifier, and as such must modify he, the subject of the main clause; otherwise it is a dangling modifier. Thus he must be the subject of the verb to meet, which remains a verb even in the gerund form meeting. Now, if the word were, say, arriving, that would be no problem; but the notion of just one person meeting is bound to strike many as odd.

As a commenter has pointed out, the problem could be solved by providing the verb meeting with a direct object:

On meeting the Estonian delegation, he was upset and worried.