A Center Controller reported an aircraft executed a missed approach to a non towered airport and did not advise ATC. The aircraft caused a NMAC with an aircraft on a Visual Approach to an adjacent non towered airport also not in contact with ATC.

Narrative

Aircraft X was cleared for the visual approach at ZZZ and was shipped to advisories approximately 15 miles out from the airport. Aircraft Y was around 10 miles behind Aircraft X on a Visual Approach into ZZZ1. I held onto Aircraft Y until Aircraft X was on short final and indicating a few hundred feet above field elevation and both aircraft were on a divergent course; opposite direction. Almost immediately after I shipped Aircraft Y to advisory frequency; Aircraft X started to climb and turn away from ZZZ airport. I tried multiple times to raise both aircraft on frequency to give a traffic alert but was unsuccessful. a min or two later; Aircraft Y reported back up to cancel his IFR; to which I called the Aircraft X traffic at his six o clock and 400 (ish) feet. He never had the Aircraft X in sight. When Aircraft X called a few mins later to report on the ground and cancel IFR; I explained the situation that was created about him going around and climbing without advising ATC. He then proceeded to tell me that he was responding to a TCAS RA on an aircraft that he observed on radar 200 ft. below him just off Runway XX. I did not observe any other aircraft in the vicinity during this event. There aren't many recommendations that I can make other than pilots do need to be aware and announce their intentions when a normal approach to an airport visually is not possible; especially when IFR. There are consistent issues within the NAS regarding what a pilot can and can't do and what they are responsible for when on a visual approach under IFR.