Secretary of State Antony J. Blinken warned on Wednesday that recent gains in getting desperately needed humanitarian aid to people in the Gaza Strip risked being undone by the fighting in southern Gaza.

The border crossing in the southern city of Rafah has been closed since Israel began what it describes as a limited military operation against Hamas fighters in the town, on the border with Egypt. The United Nations said on Wednesday that 600,000 people had fled Rafah since Israel’s ground assault started there.

“At the very time when Israel was taking important and much needed steps to improve the provision of humanitarian assistance,” Mr. Blinken said to reporters in Kyiv, Ukraine, “we’ve seen a negative impact on the fact that we have this active, very active conflict in the Rafah area.”

Josep Borrell Fontelles, the European Union’s top diplomat, said in a statement that Israel needed to end its Rafah operation “immediately,” warning that extending it “would inevitably put a heavy strain on the E.U.’s relationship with Israel.”

Israel had shut down another crossing — Kerem Shalom — after a Hamas rocket attack nearby killed four Israeli soldiers. It has since reopened, but the aid getting through is still very limited. Egypt, where most of the aid for Gaza is collected and loaded, has resisted sending trucks toward Kerem Shalom, according to multiple officials. American and Israeli officials believe that Egypt is trying to put pressure on Israel to pull back from the Rafah operation.

The Israeli military’s incursion into Rafah, which began on May 6, has created a continuing exodus of Palestinians northward, further deepening the humanitarian crisis in the enclave.