Question:
Does the United Nations have a responsibility to pressure Israel to establish a safe corridor to Egypt for those wishing to flee from Gaza?
Short Answer
Israel does not need to be pressured by the U.N to open humanitarian corridors to Egypt. Israel has long advocated for Egypt and their other Arab neighbors to accept these refugees. Egypt won't accept mass refugees from Gaza. Egypt's long standing policy is that the refugee's are Israel's making, and Israel's responsibility. This policy which unites all of Israel's neighbors is a long standing political impasse of the region's troubles.
Answer
The United Nations isn't designed to take any action not favored unanimously by its Security Council. It's primarily designed as a stage for discussions, that's it. It has no mandate as operating as some sort of world government, in the absence of that unity. There would be no unanimous endorsement by the security council to pressure Israel to do anything right now.
As for "safe corridor's to Egypt", that's a politically charged topic and ultimately not something Israel would need to be pressured to do. The issue is Egypt won't take refugee's in mass from Gaza. The Palestinian refugee's in Gaza and the West bank are displaced peoples pushed out of their homes primarily during the creation of Israel and during Israel's 1967 war. Israel's neighbors including Egypt have long held the position that the refugee's are Israel's responsibility and Israel's problem. Israel has long desired for their Arab neighbors to accept the responsibility for these refugees and that has never happened.