Konta was leading Sorana Cirstea 6-2 1-2 in the second rubber of the World Group II play-off when Nastase lost control.
As well as abusing umpire Jaume Campistol and tie referee Andreas Egli, Nastase called Konta and Anne Keothavong "f****** b******" before escorted off court and then away from Tenis Club IDU.
The International Tennis Federation has launched an investigation and removed the accreditation of Nastase, meaning he will not be able to attend the final day of the tie on Sunday.
Konta was so upset by the incident and subsequent abuse she received from the 1,700-strong crowd that, after tearfully dropping serve in the next game, play was suspended for around 20 minutes.
On the resumption, Konta reeled off five games in a row to win 6-2 6-3, tying the match at 1-1 after Simona Halep had earlier defeated Heather Watson 6-4 6-1.
But Cirstea, ranked 62, felt Konta had used the situation unfairly to her advantage and accused the 25-year-old of being too soft.
— British Tennis (@BritishTennis) April 22, 2017
"I'm not happy," Cirstea said. "Johanna, speaking tennis wise, she was better than me today. My problem was, and I still do not understand, why the match was stopped at 2-1.
"I did not hear what Ilie said, I understand you take Ilie out, perfect. Then we play another game, I win it, and then suddenly you started crying and let's go off the court.
"I've played all over the world... I never cried and left the court, I stay there and I play.
"Afterwards I go to the umpire and I ask him what do I do? I stay on the court, I go out? He said, 'I don't know'. I went to the supervisor, I said, 'What do I do, how long is the break?' He said, 'I don't know'.
"We did not exist any more, it was only the English team. And then when she (Konta) comes back on court she says, 'I'm sorry', so she knew she exaggerated.
"You don't do this, you stay and play. And the public was okay. They didn't do anything. And now they say they feel threatened.
"We are in Romania, everyone was so nice, we treated you guys so nice. I don't feel it's right that you put all this on us because we're a lower country than England. You were from the beginning expecting to jump.
"Next time I'm in trouble I will cry, maybe I can go off the court. As Romanians we get double insulted because of our nation, but it's okay, we are tough. Tougher than English people apparently."
Keothavong was particularly strong in her criticism of Nastase, who was already being investigated for asking the British captain for her room number and making an apparently racist comment about Serena Williams' unborn baby at Friday's draw.
Keothavong said: "What he said to both Johanna and myself, it's not something I am going to repeat because it's language that is not appropriate for anyone to speak to any other human in that particular way.
"It's not what we've come here for. We've come here to play a competitive match against Romania in a fair environment and we certainly don't expect verbal abuse."
Konta was left shaken by the experience, but insisted she was excited to take on world number five Halep on Sunday morning in what could be the key rubber of the tie.
The 25-year-old, speaking before Cirstea's press conference, said: "It was just unfortunate that I think the crowd took initiative from the Romanian captain's behaviour, the abusive language of everybody who was involved, it kind of snowballed.
"It has left me a little bit unnerved, but I love the sport. I am excited for the fact that the stadium was sold out, it meant that people were coming here to watch us play great tennis and that's all I'm here for. So I would just like to be able to do that tomorrow, and compete the best I can."
British number two Watson, meanwhile, may well take issue with her boyfriend, fellow tennis player Lloyd Glasspool, describing Nastase's on-court meltdown as "quality entertainment" in a tweet he later deleted.