The military said that demonstrators hurled an explosive device and several fire bombs near the fence in what it said was an apparent attempt to damage it.
Conricus told The Associated Press that the army is trying to "minimize" Palestinian casualties, but that open-fire regulations have not changed.
The protest began on March 30 and is expected to culminate on May 15.
The apologetic letter published by The Guardian was signed by four former Israeli combatants expressing their distress over the most recent killings of demonstrators in Gaza.
The ministry says 406 demonstrators suffered other types of injuries, including being hit by rubber-coated steel pellets and being overcome by tear gas.
The ministry says 203 of those hurt were treated at hospitals and the rest at field clinics.
Thousands gathered for a third consecutive week to protest for the right to return to lands their families lost in the 1948 war.
Sitting under the shade of trees near the kibbutz of Nahal Oz, he said: "Using live fire is a last resort".
Rights group Amnesty International called Friday for the Israeli authorities to "put an immediate end to the excessive and lethal force being used to suppress Palestinian demonstrations in Gaza".
"Israel is treating the protest in Gaza as it has handled similar events in the past: Broad, unlawful use of lethal force at a heavy price to lives, baseless legal interpretations issued to justify this policy, and whitewashing the crimes within days".
Smaller numbers approached the fence, throwing stones and rolling burning tires toward soldiers taking up positions on the other side. Israel says snipers only target the main "instigators".
One of the Palestinians was in critical condition, Palestinian medics said, as a protest dubbed "The Great March of Return" moved into its third week.
Denouncing Israel's "horrifying use of live ammunition against unarmed protesters", Mughrabi also noted that "under worldwide law, lethal force can only be used when unavoidable to protect against imminent threats to life".
In the northern Gaza Strip, Sumaya Abu Awad, 36, attended the protest with her three daughters and son.
Today's demonstrations took place at five tent camps located about 700 metres distance from the fence with Israel.
On Friday, rioters waved Palestinian flags and burnt hundreds of tires and Israeli flags near the border fence after Friday prayers.
The marches have been organized by Hamas, but large turnouts on two preceding Fridays were also driven by desperation among the territory's 2 million residents.
In all, 34 Palestinians have been killed in the past two weeks, 27 during protests.
Yair Lapid, the leader of Israel's centrist Yesh Atid party, called Hamas a "despicable terror organisation" and accused it of exploiting civilians.