Israeli military officials said Sunday's attacks targeted an observation post manned by militants from the Islamic Jihad organisation, Al Jazeera reported.
Defense Minister Avigdor Liberman said that the construction of the barrier constitutes an additional blow to Hamas' capabilities: "This is the only barrier of its kind in the world, which will effectively block the possibility of infiltrating into Israel by sea", he said.
The baby reportedly may not have been killed from tear gas shot by Israeli soldiers on the Palestinian protesters, part of a series of protests dubbed the "Great March of Return", but rather died from a preexisting condition.
The bombing was a response to an explosive device placed on Saturday near the border fence between Gaza and Israel, according to the army.
The deceased were identified as 25-year-old Hussein al-Amour and 28-year-old Abdul Haleem al-Naqa, the Palestinian health ministry said.
A statement from the armed wing of Islamic Jihad, the Al-Quds Brigades, claimed the two as members. Amnesty International has called the situation in Gaza "another horrific example of the Israeli military using excessive force and live ammunition in a totally deplorable way".
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Only hours before late Saturday night, Israeli warplanes carried out an air strike in southern Gaza after four Palestinians infiltrated into Israel earlier in the day.
No Israelis have been killed during that time.
Low-level demonstrations along the border have continued since protests peaked on May 14, when at least 61 Palestinians died as tens of thousands of Gazans protested the USA moving its embassy from Tel Aviv to Jerusalem.
There have been no Israeli casualties along the Gaza frontier, but Israel has reported extensive damage to farmland on its side of the border from firebomb-bearing kites flown over from Gaza.
According to Israeli media, the defence ministry ordered the construction of the barrier in the aftermath of the 2014 war with Gaza and after discovering that Hamas fighters had successfully entered Israel by sea.