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What if the conversation about my promotion gets unpleasant?
Provided you have made your request politely and respectfully, there should be no grounds for a row. If there is a row, you shouldn't be the one to start it, because you're simply asking for something, and a reasonable manager will consider your request properly.
Sadly, some bosses just don't get it, and the only way to get them to take you seriously is to resign or show them a letter from another employer offering you a job. This may be the only language they understand.
It’s also possible, however, that your boss may have fairly judged that you're not ready for the move upward, or that the right position isn't there for you. Be gracious, accept the decision, and then decide how to proceed.