Mr Morsi said he was not "calling for war", but that he would not allow Egypt's water supply to be endangered.
Egypt was apparently caught by surprise when Ethiopia started diverting the Blue Nile last month, amid works to construct a hydroelectric dam.
The river is a tributary of the Nile, on which Egypt is heavily dependent.
The Great Ethiopian Renaissance Dam is a $4.7bn (Â£3.1bn) project that Ethiopia says will eventually provide 6,000 megawatts of power.
It says the Blue Nile will be slightly diverted but will then be able to follow its natural course.
"Egypt's water security cannot be violated at all," Mr Morsi said on Monday. "As president of the state, I confirm to you that all options are open."
"If Egypt is the Nile's gift, then the Nile is a gift to Egypt,'' he said, quoting popular sayings about the river in an emotive televised speech.
"The lives of the Egyptians are connected around it... as one great people. If it diminishes by one drop then our blood is the alternative."
Analysts say Mr Morsi could be using the issue to distract attention from severe domestic political and economic challenges.
Egypt is particularly dependant on water supply from the Nile, and its growing population has been placing that supply under increasing strain.
Sudan is also reliant on Nile waters.
Egypt cites a colonial era ruling to claim a right to the majority of the Nile's waters for itself and Sudan, but Ethiopia says the ruling is outdated.
Mr Morsi said Egypt had no objection to development projects on Nile basin states, "but on condition that those projects do not affect or damage Egypt's legal and historical rights".
Last week, Egyptian politicians were inadvertently heard on live TV proposing military action over the dam.
Egyptian President Mohammed Morsi has said "all options are open" to deal with any threat to his country's water supply posed by an Ethiopian dam.