Romania coach Ilie Nastase was reportedly ordered from the court before the World Group II play-off was halted.
With Konta leading Sorana Cirstea 6-2 1-2 in the second rubber in Constanta, Nastase became embroiled in a row apparently about noise from the crowd.
The 70-year-old former world number one was reportedly heard to say to the umpire: "What's your f***ing problem?". He then appeared to call Great Britain captain Anne Keothavong and Konta "f***ing bitches".
Nastase, who could face censure over alleged racist remarks made on the eve of the game about Serena Williams' pregnancy, was asked to leave the playing arena and, after taking a seat in the stands, was then escorted back to the locker room.
Konta was visibly upset and, after playing the next game in tears and dropping serve, the players left the court.
Simona Halep addressed the crowd in a bid to ease the situation after earlier beating Great Britain's Heather Watson in straight sets to give the hosts a 1-0 lead in the tie.
Addition reporting by PA