Can water be liquefied or solidified just by adjusting the temperature, regardless of the pressure?
If I actually understood your question correctly, yes there are regions above which there will be only solid state (rather than no solid state). This is limited to the molecular region.



The Roman numerals indicate various ice phases.

From the above phase diagram (which can be found here) the solid-liquid equilibrium line doesn’t touch the y-axis as you suggest. But there will be only solid phase above the $10\ \mathrm{GPa}$ region(approximately). 

In the phase diagram you drew, if (a big if) such a PD exists for some element/molecule, then there will be no solid above the point where solid-liquid line meets y-axis. 

As pointed out in the comments, when we reach high temperatures and pressures we won’t be seeing the molecular water. It will dissociate into atoms/ions as shown in the figure.