Potters Bar Town told BBC Sport the game was postponed “following a serious incident involving a member of our management team”.
“Police attended the ground and, following their involvement, the match was unable to proceed,” the club added.
“Our manager is receiving appropriate medical care and the club is supporting him fully.
“The matter has been referred to the relevant authorities, including the league and the FA, and we are seeking formal advice. As this is now subject to ongoing processes, the club will not be making any further comment at this time.
“We thank supporters for their understanding.”
BBC Sport understands the Football Association is looking into the incident.
Announcing the postponement, Welling said: “The club will be making no further comment at this time.
“Further news regarding the rearranged fixture will be communicated in due course.”
Welling United have been contacted by BBC Sport for comment.
Nick Robinson, chair of the Isthmian League, told BBC Sport: “The league is aware of this matter and are awaiting the written reports of the alleged incident.
“We cannot comment further at this time.”
The Isthmian League is the seventh tier in English football and Potters Bar Town sit bottom of the table, with Welling two places above in 20th.
