Israel and Hamas have engaged in several bouts of fighting this month.
Israeli and Hamas officials both threatened a further intensification of hostilities.
On Facebook, Hamas spokesman Hazim Qassim slammed Israel for its "deliberate bombardment and destruction of cultural center", calling it a "barbaric act that belongs in the Dark Ages".
After the meeting a number of media outlets reported that a cease-fire had been reached. There were no instances of violence reported along the border overnight.
Israeli Prime Minister Benjamin Netanyahu was convening a meeting of his security Cabinet.
According to Israeli press reports, a total of nine Israelis have recently been injured by rockets fired from Gaza.
Tension along the Israel-Gaza border has escalated since late March, when Hamas launched what would become regular mass protests along Israel's perimeter fence with Gaza.
Brig. -Gen. (Res.) Yossi Kuperwasser, former director-general of the Ministry of Strategic Affairs, said the Israeli-Hamas flare up was a show of force by Hamas to highlight that any deal they may make with Israel was not due to any weakness on its part.
The IDF said the strike came in response to a rocket fired at Beersheva earlier in the day and was an "indication of the IDF's intelligence and operational capabilities".
Israel's military had not commented.
Abu Mujahed, spokesman for the Popular Resistance Committee, a smaller faction in Gaza, said on his Facebook page: "Upon Egyptian and worldwide efforts, a ceasefire between Palestinian resistance began conditional on the commitment of the occupation". Kuperwasser spoke with reporters through the Israel Project.
Israeli Cabinet minister for construction and housing, Yoav Gallant, said that "whatever is needed to be done to defend our civilians and soldiers, will be done, no matter what would be the price in Gaza".
"Hamas is responsible & bears the consequences for the ongoing events", the army said on Twitter. "It's a tremendous concern of ours", she said. "We are closing in on operation in Gaza", a senior IDF official told Haaretz on Thursday. The building also served as a cultural center.
The building was not publicly known to include facilities for Hamas.
Palestinians gather around a building after it was bombed by an Israeli aircraft, in Gaza City, Aug. 9, 2018.
President Reuven Rivlin said, "We do not know what is right for us and where we are going, but when there is an attack against us we will respond with force". Many residents have a reinforced room in their homes where they can shelter.
Mladenov's statement came amid the latest surge in violence this week, in which Hamas fired over 150 rockets at Israel.
"Sirens again burst the hopes of children near the Gaza border for a quiet summer vacation - no country would stand for such a situation".
On Twitter, Hamas spokesman Sami Abu Zuhri echoed similar sentiments, and called on the worldwide community to "shoulder its responsibilities towards Israel's aggression and siege".
"Israel will continue to use all necessary tools to prevent harm to its citizens and defend its sovereignty".
Hamas and Islamic Jihad, another Gaza militant group, said they had targeted Israeli communities in response to Israel's "aggression".
"These incidents underscore the urgent need to work toward finding a lasting political solution for Gaza and to respond effectively to the humanitarian crisis affecting the Palestinian population", France said.
Egypt and the United Nations have been trying to mediate a comprehensive cease-fire to prevent an escalation in fighting and to ease the deep economic hardship in Gaza, a narrow strip of land that is home to 2 million Palestinians.
British Minister for the Middle East and North Africa Alistair Burt tweeted that he was "concerned by deteriorating situation in Gaza/Israel and civilian casualties, inc. reports of death of pregnant woman and 1yr old child in Gaza".