2nd Lieutenant Hadar Goldin is thought to have been taken as a fragile ceasefire between the two sides broke down.
In the ensuing battles, at least 44 Palestinians died, while the Israeli military said that two of its soldiers were killed.
Lt Goldin, 23, thought to be one of twin brothers fighting on the frontline, was according to reports captured after gunmen emerged from one or more Gaza tunnels and opened fire at Israeli soldiers.
An Israeli army spokesman said that at least one of the militants had detonated an explosives vest.
The White House condemned the incident denouncing what it said was an "absolutely outrageous" action by Hamas.
Deputy National Security Adviser Josh Earnest called for the soldier to be released immediately.
Later in a telephone conversation the Israeli prime minister Benjamin Netanyahu told US Secretary of State John Kerry that militants had "unilaterally and grossly" violated the ceasefire.
"Israel will take all necessary steps against those who call for our destruction and perpetrate terrorism against our citizens," Mr Netanyahu told the US, according to a statement from his office.
A Hamas spokesman would neither confirm or deny the capture and said reports were being used as a cover for a "massacre".
Foreign Secretary Philip Hammond played down reports in Israeli media that the soldier had dual British-Israeli citizenship.
"We have no information to support those reports. We are looking into it but we have no information to suggest that he is a British citizen," he said.
UN Secretary-General Ban Ki-moon blamed Hamas for violating the ceasefire in Gaza as he demanded the immediate and unconditional release of the soldier.
He also called on both sides "to show maximum restraint and return to the agreed 72-hour humanitarian ceasefire that tragically lasted such a brief period".
Labour leader Ed Miliband also called Hamas a "terrible and disgusting organisation". However, he added that he thought Israel was doing itself "no favours" with its attacks on Gaza, which have killed more than 1,500 in just weeks.
For their part Israel and Hamas accused each other of breaking the ceasefire, which lasted just two hours.
The apparent capture of the Israeli soldier could lead to a major escalation of the already bloody 25-day-old conflict.
UN political chief Jeffrey Feltman said Ban is "profoundly disappointed" that the assurances from Hamas were not kept.
"Our goal was very clear: we need to end the killing," Mr Feltman said.
He said the capture of the Israeli soldier will make it more difficult to get back to the point when a ceasefire was announced.
Mr Miliband also said that David Cameron was "in the wrong place" over the Gaza conflict, and that he should be putting pressure on both sides.
"I think he is in the wrong place on this," he said, "because I agree with him about Hamas - Hamas is a terrible and disgusting organisation - but I think he should have said from the outset that this incursion by Israel into Gaza was not going to solve the problem, it's making it worse.