It also resumed its military offensive two weeks after imposing the blockade, ending a two-month ceasefire with Hamas.
Israel said the steps were intended to put pressure on the armed group to release the 58 hostages still held in Gaza, at least 20 of whom are believed to be alive.
Since the easing of the blockade, scenes of chaos have broken out at aid distribution centres run by the Gaza Humanitarian Foundation - a US- and Israeli-backed group.
The UN, which refuses to cooperate with the GHF, said 47 people were injured earlier this week after crowds overwhelmed one of the centres.
Mr Fletcher said: "We're seeing food set on the borders and not being allowed in when there is a population on the other side of the border that is starving, and we're hearing Israeli ministers say that is to put pressure on the population of Gaza."
He said that Israeli Prime Minister Benjamin Netanyahu should "absolutely" disavow a statement made by Finance Minister Bezalel Smotrich - who had said people in Gaza would be "totally despairing understanding that there's no hope and nothing to look for", and would be looking to relocate to begin a "new life in other places".
"We would expect governments all over the world to stand for international humanitarian law, the international community is very, very clear on that," Mr Fletcher said.
He called on Netanyahu to ensure that "this language, and ultimately, this policy... of forced displacement, isn't enacted".
Israel has faced growing international criticism over its conduct of the war.
On Tuesday, the EU's top diplomat, Kaja Kallas, said: "Israeli strikes in Gaza go beyond what is necessary to fight Hamas."
Her remarks followed an intervention by German Chancellor Friedrich Merz, who declared he "no longer understands" Israel's objectives.
Earlier this month, the leaders of the UK, France and Canada called on the Israeli government to "stop its military operations" and "immediately allow humanitarian aid to enter Gaza". In response, Netanyahu accused them of siding with Hamas.
On 14 May, Mr Fletcher called on the UN Security Council to act to prevent genocide in Gaza.
Asked why he had made that statement, he referred to reporting from colleagues on the ground in Gaza.
"What they're reporting is forced displacement. They're reporting starvation, they're reporting torture, and they're reporting deaths on a massive scale," he said.
Mr Fletcher said in the cases of Rwanda, Srebrenica and Sri Lanka, "the world had told us afterwards that we didn't act in time, that we didn't sound a warning".
"And that's my call to the [UN] Security Council and the world right now, 'will you act to prevent genocide?'"
Mr Fletcher came under strong criticism from Israel after he claimed 14,000 babies in Gaza would die in 48 hours if aid was not allowed into the Strip.
The Israeli Foreign Ministry accused Mr Fletcher of ignoring Hamas's atrocities and echoing their propaganda. "It's not humanitarian work, it's blood libel," the ministry said at the time.
Mr Fletcher said: "At the point when I made those comments, we were desperately trying to get that aid in.
"We were being told we couldn't get it in, and we knew that we'd probably have a couple of days, a window to get as much aid in as possible, and that was being denied, and we were desperate to get that in. And so yes, we've got to be utterly precise with our language, and we've clarified that."
Asked about his claim – repudiated by Israel - that thousands of lorries were waiting on the border to enter Gaza, Mr Fletcher repeated that he especially needed to be "careful and really precise".
He agreed there was a risk of being seen to hype the situation, but he added: "I'm not going to stop speaking up for the need to save these lives in Gaza, to save as many survivors as possible. That's my job, and I've got to do it better, and I will do it."
He said mediation and negotiation was the way to resolve the crisis in Gaza and repeated his call for Hamas to release the Israeli hostages being held by the militant group.
"We all want to see those hostages freed and back with their with their families," he said.
"I don't know now what the aim of this war is anymore. I think it has clearly gone beyond just the hostage releases. There's a lot of talk about finishing off Hamas.
"And clearly, as many people have said, there can't be a part for Hamas in the new equation, the new governance of Gaza and the Palestinian territories."
Mr Fletcher rejected Israeli claims Hamas was stealing large amounts of food aid.
"I don't want to see any of that aid getting to Hamas. That matters to us because these are our principles, neutral, impartial, independent. Its in our interest to stop that aid getting to Hamas and ensure it gets to civilians."
"As a humanitarian, my interest is just in getting as much of that aid in as possible, as quickly as possible, and saving as many lives as we're allowed to do in the time we have."
Mr Fletcher is also dealing with crises in Ukraine, Sudan and Syria, among others, and said the world was facing a "profoundly dangerous" moment.
"The Security Council is polarized, divided," he said.
"That means it makes it much harder for us to end conflicts; the conflicts we're dealing with are more ferocious, there's more impunity, and they're lasting longer.
"It's getting harder and harder to end wars and we humanitarians... deal with the consequences."
Israel launched a military campaign in Gaza in response to Hamas's cross-border attack on 7 October 2023, in which about 1,200 people were killed and 251 others were taken hostage.
At least 54,249 people have been killed in Gaza since then, including 3,986 since Israel resumed its offensive, according to the territory's Hamas-run health ministry.
With additional reporting by Olivia Lace-Evans and Maarten Lernout.
Mr Fletcher's full interview will be broadcast on the BBC News Channel at 00:30BST and 16:30BST on Saturday.