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The bill caused an outcry and Mr Kasich said it could be unconstitutional.
But he did sign a 20-week abortion ban, which is similar to a restriction already in place in 15 US states.
Mr Kasich called this bill the "best, most legally sound and sustainable approach to protecting the sanctity of human life".
The "heartbeat" bill is so called because it sets the cut-off point for legal abortions at the first time the fetal heartbeat can be detected. Such development can come before many women know they are pregnant.
Anti-abortion campaigners believe this leaves women with no choice but to continue every pregnancy.
The two bills both fell on Mr Kasich's desk at the same time and had both been approved by the Republican-controlled Ohio Senate.
The "heartbeat" bill would have been passed had Mr Kasich not used his right to veto it.
Mr Kasich said he believed that its passing would have led to costly legal challenges, which he felt the state had no chance of winning.
"The State of Ohio will be forced to pay hundreds of thousands of taxpayer dollars to cover the legal fees for the pro-choice activists' lawyers," said Mr Kasich, adding that he believed the veto was in the public interest.
Ohio lawmakers still have the option to override his veto.
Critics of the 20-week abortion ban see it as paving the way for an attempt to overrule a 1973 Supreme Court ruling, known as Roe v Wade, which states that abortion is legal until a foetus is viable, typically between 22 and 24 weeks.
President-elect Donald Trump called for a total ban on abortion during his presidential campaign, but later said he supports an exception in cases of "rape, incest and [to protect] the life of the mother".
US family-planning organisation Planned Parenthood say 99% of abortions in the US occur before 21 weeks, and the exceptions only occur in extreme circumstances.
Mr Trump has called for an end for federal funding for Planned Parenthood if the organisation continues to support abortions.