Narrow or Dam the straits of Gibraltar
In order to make the sea less salty by geological means, you need to take water out of the sea, evaporate it to dryness, then have it fall as rain.
There is a net inflow of water into the Mediterranean sea through the 14km wide Strait of Gibraltar, and evaporation causes the mediterreanean sea to accumulate salt, making it saltier than the Atlantic Ocean.
This process is however balanced by back-mixing. There is a bidirectional flow through the Strait of Gibraltar: Atlantic water flows into the Mediterranean at the surface, but saltier Mediterranean water flows back into the Atlantic in the deeps, which prevents salt accumulating in the mediterranean too much. The shallowest point of the Strait (a short distance west of the narrowest point)is the Camarinal Sill at 280m deep.
It is theorized that it the past, the Strait of Gibraltar was indeed closed, and the mediterranean dried up, becoming hypersaline as a result, causing the Messinian Salinity Crisis from 5.96 to 5.33 million years ago. This ended when the Strait reopened in Zanclean Flood.
If you can restrict the Strait of Gibraltar such that inflow can occur but backmixing cannot, then the Mediterranean will become more salty and the rest of the worlds oceans will correspondingly reduce.
The chances of the Strait being restricted by just the right amount for the length of time to allow for the mediterranean to be completely filled with salt deposits (rather than simply closing off, drying out and becoming a land depression) naturally are not that high (but see edit!) According to the Wikipedia article there were proposals in the early 20th century to build a hydroelectric dam on the strait. The engineering challenges would be enormous, as would the electric output. There would be major environmental consequences, including climate change in Europe and Africa.
I suppose if you didn't want to build a pressure-withstanding dam, you could just fill the strait with rubble, which might very effectively prevent the backmixing while allowing the inflow. But I can't think of an economic motive for doing this, or a natural process that would achieve the same effect.
How long would this take, and how much salt could be removed?
According to https://www.mathscinotes.com/2015/08/drying-up-the-mediterranean/ it would take 1000 years for the mediterranean to dry up with no inflow. Sea water contains about 35g salt / kg water (about 30:1 by mass) so if the mediterranean were replenished through a Gibraltar Dam instead of being allowed to dry out, the mass of salt in the mediterranean would equal the current mass of water in 30000 years. The density of salt is higher than water, so there would be room for quite a bit more.
The mediterranean sea has a volume of 3,750,000km3, the world's oceans have a volume of 1,350,000,000km3, so the mediterranean is about 1/300 times that of the oceans. As noted above, about 1/30 of sea water by mass is salt, so the water in the mediterranean weighs as much as 1/10 of the salt in the oceans. As salt is denser than water you could get quite a bit more than 1/10 of the ocean's salt into the mediterranean (assuming a density ratio of 2 it would be 1/5.) That's a bit less than you were looking for but in the same order of magnitude.
EDIT: on rereading the wikipedia article on the Messinian Salinity Crisis, it seems that the straits of Gibraltar opened and closed multiple times during the Messinian period, with the Mediterranean basin drying out and filling up again repeatedly. The estimated salt deposits reached over 1 million cubic kilometres, or about a quarter of the total basin volume, and up to 4x10^18kg by mass (taken together this would imply a density of 4 times that of water for the salt, which is too high, so the actual figure would presumably have been between the two.) Based on the 4x10^18kg figure, this would have caused a drop in salinity in the world's oceans of around 3g salt per kg, or about one 1/11th of current average salinity. Geologists are saying this actually happened.