In our culture we generally ask people to know that whether they have eaten or not. There is no intention to offer food to them it's just as casual as saying hello to begin the conversation. Example we ask "did you have your food? If they had it, they say Yes I had, I just finished my lunch etc. What are the expressions that native speakers use to know that some one has eaten his or her breakfast, lunch or supper?
I expect this kind of greeting is a cultural artifact. In most of the English-speaking world, we ask how you are, not whether you have eaten.
Americans (and, I assume, the British and other English-speaking nationalities) do not casually ask if someone has eaten, as it is normally considered an invitation, and usually includes some offer of food or drink. For example, suppose John has come to visit Alan, who he doesn't know very well. Alan wants to seem hospitable, so he offers John some food:
Alan: Hello John, please come in. It's good of you to drop by. Are you hungry? We can order up a pizza ...?
John: No, no thank you. I just had lunch, but I appreciate the offer.
Alan: Something to drink then? Soda? Beer? Wine?
John: Thanks, I'll take a beer if you're having one.
Side note: In a similar way to the way your culture asks "Have you eaten", in the English-speaking world "How are you?" is rarely an invitation to talk about your actual condition. It's just a polite salutation. The expected response is, "I'm good, how are you?" even if you are not well.
The phrase "Have you eaten?" is used, but not necessarily as an indirect conversation starter. You would ask this more out of politeness.
For example, if I had a guest visit my house around lunch time I might ask "Have you eaten?" early on in the conversation and would attempt to offer them food if they replied that they had not.
It's not a must to ask this and is not a formal salutation, it's just being polite.