Romania's captain was unhappy after his comments were reported about Williams' baby, which is due in the autumn. "Let's see what colour it has. Chocolate with milk?," he was heard to say on Friday.
On Saturday morning - the opening day of Romania's Fed Cup play-off against Great Britain - Nastase stormed into the press centre at the Tenis Club IDU in Constana to seek out the British media.
Press Association's tennis correspondent Eleanor Crooks was the only member of the British media present in the press room at the time and reported that he said the following:
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Why did you write that? You're stupid, you're stupid.
The tirade took place in front of a number of members from the Romanian media and, according to Ms Crooks, lasted between two and three minutes.
Later, Nastase was abusive again towards Crooks as he was escorted away by security, calling her both "stupid" and "ugly".
Press Association has sent details of Nastase's outburst to the International Tennis Federation.
Ms Crooks said: "He repeatedly called me stupid, asked me why what he said was racist.
I explained we simply reported what he said and that it was unnecessary to make such a comment about colour. He said the English were out to get him and called me stupid a few more times.
"Fortunately he was across the other side of the room from me and there were other journalists around so it was unpleasant rather than threatening.
"But it is certainly not the behaviour you would expect of someone in his position and wholly unnecessary, especially given he did not dispute the accuracy of what was reported."

Tennis correspondent Eleanor Crooks said the incident was "unpleasant"

Image credit: PA Sport

Williams announced on Wednesday she is expecting her first child with fiancé Alexis Ohanian, who is white.
Nastase was not asked about Williams himself on Friday but made the comment, in Romanian and clearly audible, to the other players while Romanian number one Simona Halep was answering a question about the 23-time grand slam singles champion.
The ITF launched an investigation into those comments and will be under pressure to take further action due to Nastase's behaviour on Saturday.
It is the second controversy caused by Nastase, who was appointed Fed Cup captain in October, about Williams in a month.
In an interview with Romanian media, the former world number one made totally unsubstantiated allegations of doping by Williams based purely on her physique.
The 70-year-old, who had a reputation as a playboy during and after his playing career and a former president of the country's tennis federation, also clearly made Great Britain captain Anne Keothavong feel uncomfortable with a number of inappropriate remarks.
He asked for her room number during the official dinner on Thursday and then repeated the request when the captains posed together for photographs following Friday's draw while putting his arm tightly around Keothavong's shoulder.
When both teams were called back together, Nastase said to Keothavong, who is married and 18 weeks pregnant with her second child: ''We keep being attracted.''
The Fed Cup tie between Romania and Great Britain got under way on Saturday with Halep taking on Heather Watson, with Johanna Konta then set to face Irina-Camelia Begu.
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