Estimates suggest there has been at least a doubling in requests in Brazil and an increase of a third in other countries.
Many governments have advised women not to get pregnant due to the risk of babies being born with tiny brains.
The findings were published in the New England Journal of Medicine.
A termination remains illegal in many parts of Latin America, but women simply turn to unofficial providers.
Women on Web, which advises women online and then delivers pills to end a pregnancy, is one of the largest.
The researchers analysed the thousands of requests received by Women on Web in the five years before the Pan American Health Organization issued its warning on Zika on 17 November 2015.
It used this to predict how many abortion requests would have been expected between 17 November 2015 and 1 March 2016.
The analysis of countries that advised against getting pregnant suggested Brazil and Ecuador had had more than twice the expected demand for abortions.
Analysis from other countries, which did not advise against pregnancy, suggested smaller increases in abortion demand.
One woman from Peru told Women on Web: "I'm very concerned, I'm two months pregnant and in my country Zika has been detected.
"We are all very alarmed and I do not want have a sick baby, please, I do not want to continue my pregnancy because it is very dangerous."
Another from Venezuela said: "I contracted Zika four days ago.
"I love children, but I don't believe it is a wise decision to keep a baby who will suffer. I need an abortion. I don't know who to turn to. Please help me ASAP."
Dr Catherine Aiken, one of the researchers, from the University of Cambridge, told the BBC News website: "Everywhere governments said, 'Don't get pregnant' and there was Zika transmission, there was a tremendous surge in the number of women taking matters into their own hands.
"There were huge increases in abortions across the region."
Dr Aiken criticised the countries' "very hollow" messages to delay pregnancy that had generated "fear, anxiety and panic with no means to act on it".
Meanwhile Abigail Aiken, an assistant professor from the University of Texas at Austin, said: "Accurate data on the choices pregnant women make in Latin America is hard to obtain.
"If anything, our approach may underestimate the impact of health warning on requests for abortion, as many women may have used an unsafe method or visited local underground providers."
Prof Jimmy Whitworth, from the London School of Hygiene and Tropical Medicine, said the report "agrees with what I have heard informally from several sources in Latin America about increased interest in finding out more, and in making requests for abortions".
He told the BBC: "This apparent increase in making requests for abortion looks plausible and is not surprising given the situation with the epidemic and societal pressures."