The Grand Ethiopian Renaissance Dam has been a source of tension in the Nile River basin ever since Ethiopia broke ground on the project in 2011.
Addis Ababa says the dam is crucial for its economy, while Cairo fears it will disrupt the river that provides almost all its water.
Talks earlier this year involving the two governments and Sudan -- another downstream country -- failed to produce a breakthrough.
In April Ethiopian Prime Minister Abiy Ahmed proposed proceeding with "first stage filling" that would collect 18.4 billion cubic metres of water in the dam's reservoir over two years.
But both Egypt and Sudan fear the reservoir -- which has a capacity of 74 billion cubic metres -- will trap their essential water supplies.
Filling and operating the dam "would jeopardize the water security, food security, and indeed, the very existence of over 100 million Egyptians, who are entirely dependent on the Nile River for their livelihood," Egyptian Foreign Minister Sameh Shoukry said in a letter to the UN Security Council dated May 1.
"This is a situation that potentially poses a serious threat to peace and security throughout the region," he said.
In a response dated May 14 and seen Monday by AFP, Ethiopian Foreign Minister Gedu Andargachew accused Egypt of being obstructionist.
"Ethiopia does not have a legal obligation to seek approval of Egypt to fill the dam," Gedu said.
"Ethiopia has made extraordinary efforts to accommodate Egypt's unending demands and unpredictable behavior," he added.
Egypt wants Ethiopia to endorse a draft agreement emerging from the talks earlier this year facilitated by the US Treasury Department, which stepped in after Egyptian President Abdel Fattah al-Sisi put in a request to his ally US President Donald Trump.
But Ethiopia skipped the most recent round of those talks and denies any deal was agreed.
While Egypt's letter to the Security Council raises the stakes further, the possibility of armed conflict stemming from the dam dispute is still "very unlikely", said William Davison of the International Crisis Group, a conflict-prevention organisation.
"We could expect some sort of diplomatic escalation, more aggressive rhetoric. But a negotiated resolution to this is obviously the best way out for everyone, and there still seems to be plenty of possibility of that," Davison said.