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From the following sentences, which one is more grammatically correct?
I personally think the second one should be correct because we are comparing carbohydrates to fats and proteins.
Neither is quite right. In general, you would probably want to say either
Carbohydrates in the diet need less water in comparison to proteins and fats.
or
Carbohydrates in the diet need a smaller amount of water in comparison to proteins and fats.
At a stretch, you might possibly say
Carbohydrates in the diet need a lesser amount of water in comparison to proteins and fats.
but it sounds less idiomatic than the other two options.
You've hit upon a rather tricky part of English, the usage of less vs lesser vs smaller. In general, you can compare uncountable or mass nouns (like water) using the word less:
There is less milk in my glass than in yours. The water is less than one inch deep.
You use smaller to compare countable nouns:
My glass is smaller than yours. I want the smaller cookie, please.
In your sentence, you have the countable, compound noun [an] amount of water. Because it is countable, you need to have an article in there, and you need to use the comparative term smaller:
Carbohydrates in the1 diet need a smaller amount of water in comparison to proteins and fats.
Alternatively, you could drop the phrase [an] amount and have just the mass/uncountable noun water (which doesn't need an article), in which case you would use the term less for comparison:
Carbohydrates in the diet need less a̶m̶o̶u̶n̶t̶ ̶o̶f water in comparison to proteins and fats.
So why not lesser?
Lesser is a more specialized case. For one thing, it can only be used attributively (right before a noun), and only with count nouns. When we use it comparatively, it most often refers to quality or importance rather than quantity:
Joe is a lesser man than Pete.
Titus Andronicus is often regarded as one of Shakespeare's lesser works.
It can be used to refer to quantity, but only with a few specialized count nouns of amount or quantity:
I like raspberries and, to a lesser degree, strawberries.
There is a lesser amount of money invested in Plan A than in Plan B.
A lesser number of citizens are voting nowadays than in years past.
However, while all of these are to some extent grammatical, only the first sounds fully idiomatic to my ear; for the rest, I would prefer to replace lesser with smaller. An Ngram comparing usage suggests this trend is widespread:
1 The word diet is countable, so it needs an article here. You could also use a possessive pronoun: Carbohydrates in your diet.
So while you could use it, smaller probably sounds better.
Carbohydrates in diet need less amount of water in comparison to proteins and fats.
Carbohydrates in diet need lesser amount of water in comparison to proteins and fats.
Neither one is grammatically correct since diet is a countable noun and you're not comparing carbs to proteins and fats taken together. In comparison to is also needlessly verbose, leaving aside that it should really be with. The grammatically correct phrasing would be
Dietary carbohydrates need less water than proteins or fats.
Dietary carbohydrates need a lesser amount of water than proteins or fats.
Even then, neither sentence really makes sense. You should explain to the reader: ...need less water to/for what?
The use of less or lesser can be used in this sentence with minor grammatical changes.
Use of Less
Carbohydrates in diet need less water in comparison to proteins and fats
Use of Lesser
Carbohydrates in diet need a lesser amount of water in comparison to proteins and fats
In the context, the second option would be better.
