The groundwater beneath the nuclear plant, which does not contribute to drinking water, flows into the Hudson River at a point about 25 miles north of New York City.
Indian Point's parent company Entergy said elevated levels of tritium, a radioactive isotope of hydrogen, had been detected in the groundwater at the facility, but that "there is no health or safety consequence to the public."
The leak also does not pose a threat to the river environment because dilution would render the radioactive water "undetectable," said Neil Sheehan, a spokesman for the Nuclear Regulatory Commission.
On Saturday, Cuomo said he was "deeply concerned" by the leak and directed the heads of the state departments of health and environmental conservation to investigate.
"Our first concern is for the health and safety of the residents close to the facility and ensuring the groundwater leak does not pose a threat," Cuomo said.
There are 317,000 residents within a 10-mile evacuation zone around the nuclear facility, according to the Natural Resources Defense Council.
The amount of released contaminant is far below federal permissible limits and the leak was reported to the Nuclear Regulatory Commission and state agencies voluntarily, Entergy said.
"The more immediate concern is how did this happen?" Sheehan said.
Entergy has launched an investigation into the incident and the NRC plans to send an expert in health physics and radiation protection to the site next week, Sheehan said.
In a letter to the commissioners, Cuomo said, "We need to identify whether this incident could have been avoided by exercising reasonable care."
According to Cuomo's office, three out of 40 monitoring wells reported "alarming levels of radioactivity," with one well detecting over 8 million picocuries per liter of radioactivity (the standard measure caused by the tritium, up from a standard 12,300 picocuries per liter).
The radioactivity levels recorded in the groundwater are the highest ever detected at Indian Point, Sheehan said.
The likely cause of the elevated radioactivity detected was a "spillage of water as a result of a mechanical issue during pumping of water," last month, according to Entergy spokesman Jerry Nappi.
An "out-of-service" sump pump caused the contaminated water to overflow a containment drain and leak out of the building, eventually seeping into the groundwater at the site, NRC spokesman Sheehan said.
The elevated levels of radioactivity were first noted earlier this week after results from routine testing of the special groundwater monitoring wells at the facility.
Cuomo and Sheehan both said they were notified of the radioactive leak on Friday.
In its statement, Entergy said, "The issue of tritium in the ground is a well-studied issue, including by the state Department of Environmental Conservation, which has previously concluded historic leaks of tritium to the ground at Indian Point have no effect on public health or safety."