The IVF pioneer said he had been kicked by the woman when he stopped her as she was cycling through Bloomsbury, central London, on Wednesday.
She became "abusive" and tried to snatch his phone after he pointed out she was breaking the law, he said.
"I went up to her and told her very politely that it was against the law to cycle on the pavement and it was dangerous," he told The Times.
"I thought she would apologise and walk off but she became very aggressive and was swearing... She then kicked me repeatedly."
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Two passers-by came to his aid, he added.
The Labour peer told the paper he had not reported the incident to the police because it would be "impossible to identify the cyclist".
He said he was now considering introducing a private members' bill to force cyclists to have licence plates.
"The requirement for number plates would mean cyclists who are blatantly breaking the law can be identified," he said, adding that he believed it could help prevent "aggressiveness" shown by some cyclists.
His call comes just days after he and other peers raised concerns at Westminster over the irresponsible behaviour of some aggressive bike riders who ignored red lights and pedestrian crossings and rode on pavements.
Lord Winston told the House of Lords: "Most cyclists are conscientious and law-abiding but an increasing number are extremely aggressive."