Not exactly, noise level is inversely correlated with the photosensitive element size. The smaller they are, the more noise you have.
For example an 8MPX APC sensor should have less noise than a 40MPX full frame sensor.
Per-pixel noise depends indeed on the pixel size, but image-scale noise is almost completely independent of the pixel size.
That's because while a large pixel has less noise, it appears larger in the output and so the noise is correspondingly more visible. A small pixel is noisier, but smaller, so the net noise is the same for the same sensor size.