Researchers have estimated an increase in double the amount of abortion requests in Brazil and a third in other Latin America countries.
Governments have advised women to avoid pregnancy while the risk is high but at times this is unavoidable.
The findings were published in the New England Journal of Medicine.
Nearly sixty countires have found cases of the Zika virus with more than more than 1,500 babies having been diagnosed with microcephaly.
In some parts of Latin America abortion remains illegal so women are resorting to illegal and unsafe procedures.
“Women on Web”, a service that delivers abortion pills to terminate a pregnancy, is the largest of these services.
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 were analysed to draw this data.
It used this to forecast how many abortion appeals would have been anticipated between 17 November 2015 and 1 March 2016.
The examination of nations that warned against getting pregnant saw that Brazil and Ecuador had had more than twice the estimated amount.
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 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".