Short: No, you can't. ;)
As already mentioned "sauer" means "sour" so "italienisches Sauer essen" could be roughly translated to "Italian sour food" although the term itself is grammatically not correct.
The term "Sauerkraut" literally translates to "sour cabbage". That is because it is cabbage which is conserved through fermentation. The bacteria growing during the fermentation produce acid making the cabbage taste sour.
In Europe, and especially not in Germany, the terms "Sauer" or "Kraut" are not used to refer to food in general.
You might find German speaking people talking about "Kraut" as an abbreviation for "Sauerkraut"
but this only refers to Sauerkraut..
edit: as mentioned by @IQV this is not correct. "Kraut" may also refer to other mostly cabbage-based dishes. The definition of what "Kraut" actually means depends a lot on the dialect spoken in that geographical region.
Ja, warum nicht, wir könnten vielleicht italienisches Essen essen.
is grammatically perfectly correct. But it might refer to just eating Italian food at home, somewhere else or at a restaurant offering several kinds of food and does not explicitly express that you wish to do so at an Italian restaurant.
If you want to express that you want to go to an Italian restaurant the term
Ja, warum nicht. Lass uns doch zum Italiener gehen.
is what you are looking for.
In German everyday language restaurants are mostly referred to by naming a male individual of the restaurant's nationality.
A Greek restaurant - Ein Grieche ("Lass uns zum Griechen gehen")
An Italian restaurant - Ein Italiener ("Lass uns zum Italiener gehen")
An Indian restaurant - Ein Inder ("Lass uns zum Inder gehen")
And so on.
One exception (and the only one I can think of right now) is if you are going to a German restaurant in Germany.
Restaurants serving German everyday food are mostly referred to as "gutbürgerlich".
In this case you normally don't talk about going to a German restaurant ("... in ein deutsches Restaurant gehen.") but about going to a plain restaurant ("... in ein gutbürgerliches Restaurant.")
edit: @O. R. Mapper has a good point regarding the restaurants serving German cuisine: Restaurants serving German food are actually rarely called "gutbürgerlich" in German everyday language. Instead one might refer to them with the kind of cuisine they serve (vegetarian, vegan ...) or the region the cuisine of the restaurant originates from (bavarian, bohemian, swabian, ...)
Her you can also refer to the male individual representing that kind of cuisine (the Vegetarian, the Bavarian, the Swabian) but this is very informal and depends a lot on the regional dialect.
For example you might say
Lass uns in in vegetarisches Restaurant gehen - Let's go to a vegetarian restaurant.
Lass uns in in böhmisches Restaurant gehen - Let's go to a bohemian restaurant.
and so on.
This is the most common option and you should use such phrases.
You MIGHT hear
Lass uns zum Vegetarier gehen - Let's go to the Vegetarian.
Lass uns zum Böhmen gehen - Let's go to the Bohemian.
and so on as well.
But as I already said this is very informal and depends a lot on the regional dialect and therefore I do not recommend to use these phrases. I only mention this for the sake of completeness.