Quite often, and this is something I've run into a lot myself, it's not about how you perceive yourself or your actions, but how the other person perceives you. While we shouldn't let what other people might think stop us from doing something, that doesn't mean we should be oblivious to our impact on others. We can only continue to relate to a person if we understand how they feel. If we're not willing to accept that this is how they felt, and potentially do something about it, then we've come to an impasse on interacting with this person. Rather than try to understand how they reached they conclusion, and why, we've essentially said their feelings are invalid and irrelevant, which in turn very likely says they are irrelevant.
Even something as simple as "I may not agree with you, but I understand how you reached that conclusion" would go a long way toward bridging differences, but additionally "in the future I'll be [xyz (e.g. more explicit, more attentive, etc)]" may be the difference between parting ways, or a luke-warm acquaintance, instead of close ties and better understanding.
If you were to ask any of my previous co-workers or managers you would find this is something I've struggled with quite a bit myself, so I understand what it means to be in that place, and have learned to accept that regardless of how I may think I'm being, the end result may be far from it (especially when it comes to text-only communications).