In Gaza's southern Rafah area, the military urged residents in phone calls to stay indoors as troops advanced.
"We are under fire. Every minute or so, tanks fire shells," said Ayman al-Arja, 45, a resident of the area.
Meanwhile, U.N. Secretary-General Ban Ki-moon blamed Hamas for violating what was meant to be a three-day humanitarian cease-fire and demanded the immediate and unconditional release of the missing soldier.
Both Israel and Hamas accused each other of breaking the cease-fire, which had been announced by the U.S. and the U.N. and took effect at 8:00 a.m. Friday.
The breakdown of the truce and the apparent capture of the Israeli soldier set the stage for a major escalation. The conflict has already devastated large swaths of the coastal area and killed at least 1,600 Palestinians, mainly civilians, according to Palestinian officials. Israel has lost 63 soldiers and three civilians.
The Palestinian death toll rose by 160 on Friday, including 140 who were killed or died of injuries and 20 bodies recovered from the locations of Israeli strikes, Gaza health officials said.
The clash between Israeli troops and Hamas near the town of Rafah began about an hour after Friday's cease-fire started.
Gunmen emerged from one or more Gaza tunnels and opened fire at Israeli soldier, with at least one of the militants detonating an explosives vest, said Israeli army spokesman Lt. Col. Peter Lerner.
Hadar Goldin, a 23-year-old second lieutenant from the central Israeli town of Kfar Saba, was apparently captured during the ensuing mayhem, while another two soldiers were killed.
"We suspect that he has been kidnapped," Lerner said.
The White House condemned the incident, describing it as an "absolutely outrageous" action by Hamas. Deputy National Security Adviser Josh Earnest said the soldier must be released immediately.
Israeli Prime Minister Benjamin Netanyahu, meanwhile, told U.S. Secretary of State John Kerry in a telephone conversation that Palestinian militants had "unilaterally and grossly" violated the ceasefire and attacked Israeli soldiers after 9 a.m.
"Israel will take all necessary steps against those who call for our destruction and perpetrate terrorism against our citizens," Netanyahu told Kerry, according to a statement from the prime minister's office.
Mark Regev, Netanyahu's spokesman, said Hamas had "yet again thrown away a chance for a humanitarian relief for the people of Gaza, by deliberately violating this ceasefire."
Moussa Abu Marzouk, Hamas' deputy leader, told Al-Arabiya news channel from Cairo that the movement's military wing carried no military operations after 8 a.m., when the truce came into force.
If confirmed, Goldin's capture could dramatically change the trajectory of the conflict. Any cease-fire efforts would likely be put on hold and Israel might instead expand its ground operation. Israel has in the past gone to great lengths to return captured soldiers. In 2011, it traded hundreds of Palestinian prisoners for an Israeli soldier who had been captured by Hamas-allied militants in 2006.