It can be quite tricky to know whether it is okay to ask about a topic like this. On the one hand, I have met several war veterans who like to talk about their experiences because they like to reminisce about their younger days or as a warning to younger generations. On the other hand, I've also met war veterans who don't want to talk about their experiences because of the trauma or because they are ashamed to have been part of such violence. When I encounter a situation like this, where I want to ask about something specific, but don't know if it okay to do so, I like to ask questions tangential to the topic.
Asking a question that is tangential to the topic allows you to open the door for them to talk about it, without directly bringing it up. If the other person is not willing to talk about the subject, they have the option to avoid it and there is no awkward encounter of them declining to answer your question. If they are willing to talk about the subject, you've given them an avenue to do so. I've been using this method for several years to bring up subjects that I wasn't sure it was okay to talk about. One particular example is a date that I went on a few years ago. It was going well and I was interested in seeing if the night would lead to sex. I didn't want to just straight up ask if she wanted the same thing, because I didn't want to make things awkward and ruin the date if she wasn't interested. I don't remember exactly what I asked her, but it was something akin to asking what she wanted to do after dinner. At that point she was able to suggest that we go back to my place and, after a few more well placed questions, she ultimately initiated the sex. I've been in similar situations as well, where the person I was with instead suggested going for a walk, going to get dessert, etc... and it wasn't awkward.
How you should approach this
When I was in high school, I did a lot of volunteer work with elderly residents in a nursing home. What I found was that they liked having someone to talk to, and they would talk freely about anything they felt like talking about. There are some good related questions you can ask to open the door for the man you've met to talk about being in the war. I recommend that you focus on topics related to when he was young. Ask about what he did as a young man, who his friends were, etc... As he answers these questions, you should get even more information that you can use to ask more specific questions to steer the conversation towards the war, but don't actually go directly there. Wait for him to bring it up, and when he does you can start asking about it.