The plates float and oceans are deeper than continents because the oceanic plates are heavier. There is not enough pressure to raise the cut off piece of the plate to the surface, it is too heavy. Any pressure would just come out at the edges that had been cut.
But there is a way to create new countries, kind of. Nuclear bombs can be made directional (the US even developed some for use as EMP weapons), and AFAIK there is no real upper limit to the size of thermonuclear bomb. So you could make a bomb that can cut a hole down to sufficient amount of magma. And if you made enough of the bombs you could get enough of the lava to get your small country. So it is just a matter of money for a nuclear power.
Ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn
There are just some minor issues, though. First, exploding large amount of nuclear devices of unprecedented power would not make people happy with you. Thermonuclear bombs could be made relatively clean, but there is no way to get rid of the shockwaves, electromagnetic pulses, tsunamis... So you'd become pretty unpopular very fast.
Not that it really matters because as soon as the magma started flowing all that would be forgotten. Feel free to read the Wikipedia article on Large Igneous Province as that is essentially what you are trying to create. Pay attention to the part there they list all the past extinction events that have been linked with fast creation of large amounts of land with volcanism. Including the one that is also called "the great dying" because it made the majority of everything go extinct. Of course every single hole created would also qualify as a massive explosive eruption.
So this would only make sense if you do not care about every single person on the planet dying, large number of species going extinct, and triggering a new ice age. Probably a short one, but still. Seriously the nuclear winter people used to worry about would have been child's play. And while you can probably come up with a way to trigger the eruptions that does not involve ridiculous number of nuclear explosions, you can't really avoid the consequences of enough magma to raise a small country erupting when your goal is to create a small country from erupting magma.