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era ruling to claim a right to the majority of the Nile's waters for itself and Sudan, but Ethiopia says the ruling is outdated.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.

Romelu Lukaku has joined Manchester United for £75m, Arsenal have snapped up France forward Alexandre Lacazette for a club-record £46.5m, while Everton have already topped the £100m mark for money spent.
But how much are those stars on the move really worth? And what fees will others linked with big-money transfers command?
The CIES Football Observatory has calculated the value of the world's top players using variables such as performance, international status, contract, age and position.
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The Whites will move above Sheffield Wednesday into the top six if they beat Blackburn at Craven Cottage on Tuesday.
Button, 28, who joined Fulham last summer from Brentford, told BBC London: "There's 10 games to go, nearly a quarter of a season.
"A lot of things can change so as confident as we are, we've got to make sure we maintain that."
Fulham are the Championship's in-form side, taking 17 points from their last seven games, including a 3-1 victory at leaders Newcastle on Saturday.
Reaching the play-offs would mark a dramatic turnaround in 15 months under head coach Slavisa Jokanovic after they finished 20th last season.
The former Watford manager transformed the squad ahead of his first full season, including bringing in Button - and the goalkeeper praised Jokanovic for his impact.
"The gaffer and his team know exactly what they want from us and drilled it into us from day one," he added.
"Why it's clicked recently I'm not too sure. Obviously there were a lot of changes in the summer, a lot of new faces.
"We're getting to know each other and we've shown on the pitch that it has come together."
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.
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We're less than a fortnight into the summer transfer window, but Europe's top clubs are already flexing their financial muscle.
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Goalkeeper David Button says Fulham can not afford to be complacent as they target a Championship play-off place.