It is indeed the case that Ray of Frost does 1d8 damage, while a comparable weapon might do 1d8+ability modifier damage. However, since you can use your spellcasting stat (assumed to be your highest ability), you will have a better chance of succeeding in your attack roll. In addition, Ray of Frost slows on a hit, which is a powerful additional effect.
I didn't find any rules text that states "Don't use the spell description but some other rule similar to weapon attacks", but for reference, here's an excerpt of the section about weapon damage rolls in the How to Play:
If you’re attacking with a melee weapon, apply your Strength modifier to the damage, and if you’re attacking with a ranged weapon, apply your Dexterity modifier.
And Ray of Frost, from the final playtest:
A beam of blue-white light streaks toward a creature within range. Make a ranged attack roll against that creature, with a bonus to the roll equal to your magic ability modifier and your spellcasting bonus, if any. On a hit, the target takes 1d8 cold damage, and its speed is reduced by 10 feet until your next turn.
As nitsua60 noted in a comment, there are certain class features that allow a caster to add her or his spellcasting modifier to a cantrip, e.g. the Warlock's Agonizing Blast. If any caster could just add the modifier, this feature would be pointless, or would at least be worded differently (add double your modifier or so...).
So yes, the ray does a bit less damage, but with the control effect it is equally good if not stronger than a 1d8+3 weapon attack - and you can use your primary stat for attack rolls, which could easily be a +1 or +2 on the to-hit.
As written, it does not affect the versatile damage.
The description of versatile says (PHB p.147):
Versatile. This weapon can be used with one or two
hands. A damage value in parentheses appears with the
property—the damage when the weapon is used with
two hands to make a melee attack.
The spell does exactly and only what it says "the weapon's damage die becomes a d8". So a quarterstaff does 1d8 damage 1 handed and remains a versatile weapon doing ... 1d8 damage when wielded 2 handed.
Is it "feasible" to say it increases the damage? No. Is it "reasonable" to do so? Why not, its not going to break anything and it makes enough sense that I'm not going to come around to your house and tell you "You're playing it wrong!"
Best Answer
Yes.
Weapon sizes are addressed in the Creating a Monster section of the DMG. On p 278 it says:
While this is different from the "Heavy" notation used in the PHB it is not inconsistent if we assume that "Heavy" weapons are sized for Medium and the others are for Small.