um. without reading you post and just reading your description i was already shaking my head in disapproval.

so no. i dont think a relationship can last without communication. 

relationships are communication. yes idk what you guys were fighting about, so idk if you were right or wrong, but you guys still need to talk about whats going on. if you are uncomfortable with something and want to talk about it fix it, your SO needs to let you, even if it is hard for them to talk about it.

if your SO thinks youre annoying by wanting to talk, then maybe you guys need to talk about: why you feel the need to talk, and why the other is so stifled by the need to talk about it. the fact that there was something that you guys were fighting about means that there was already a miscommunication taking place, and the fact that your SO further refused to sort it out is just conformation that they arent taking you/ your relationship as seriously as you are. 

if you want to talk, your SO at least needs to hear you out. its a two way street, you guys aren't going to agree on everything (obviously) so you need to communicate to work things out and be on the same page.

to me, no communication/ not being good at communicating/ actively avoiding communication are signs that the relationship is not serious/ going to last.

dont blame yourself for this, you wanted to work things out.