I am not diagnosed with depression, but I believe I have it. Could I still get admitted to a mental hospital for suicidal thoughts?

It mostly depends on the process where you live. I would think you wouldn't be admitted just for having thoughts alone, unless they believe you had immediate plans of action or were in any danger. Passive suicidal thoughts are common and you do deserve to receive support for them, as well as for any depressive symptoms you may have, regardless of whether you have a diagnosis of depression. If you were afraid you may be admitted for talking about your thoughts, I think it's not something you should worry about, and that it would be worth it for you to reach out. If being admitted was something you feel you need, it's also something that you could bring up with someone, to discuss if that's the right option for you. Either way, reaching out to someone, such as a friend, family member, or doctor (even a family doctor), could be beneficial, so that you don't have to deal with these thoughts alone.