No, it shouldn't, because obviously it doesn't.
However, if we observe ONLY the pattern of molecular weight, then yes it "should".
Heavier molecules on average have lower melting and boiling points. Because heavier, larger molecules have stronger bonds between them, and they are harder to get going, so the bonds won't be broken that easily and you need a larger temperature to get a change in state as opposed to a lighter molecule.
Water is pretty light. It is basically methane. Methane boils at -160 °C.
Water boils at much higher temperatures, why? Because molecular weights are in fact not the only thing affecting the properties of matter.
When elements bond together covalently they share their electrons. But not all of the electrons get shared. Due to the configuration of oxygen and hydrogen, there are actually 2 pairs of valence electrons on the O atom that aren't taken up by the H of the same molecule, so they hang on the side.
This allows H atoms of neighboring molecules to be attracted to them, and relatively strong _hydrogen bonds_ are formed. One water molecule is connected to 4 other ones in normal conditions. This is why water molecules stick together tight and won't come loose at temperatures as low as e.g. methane.