Why would we?
To make room for the increasing population around the Mediterranean Sea?
But what do you think attracts so many people around the Med?
In 3 S words: Sea, Sex, and Sun.
If you remove the Sea, who would consider living by a northern Sahara Dead Sea?
Why can't we?
The easiest way would be to build a dam across The Strait of Gibraltar.
So the Mediterranean would slowly evaporate into a smaller Dead Sea-like lake 200 m below (in Mediterranea, the evaporation exceeds water intake from all rivers).
It should be a 15 km long, 900 m deep intercontinental dam (or 35 km / 250 m if further east) in an unstable area where African tectonic plate is bumping into European at 1 cm/year.
I let you imagine a breach in that dam!
A disaster similar to the Black Sea event 5500 BC which probably inspired Flood accounts among surrounding populations.
Huge cost, huge risk, no real needs, no ROI, insufficient technology...
An interesting case study, certainly not a project.