I agree with Bonzi. It’s totally normal to have awkward pauses in conversation. Furthermore, why awkward? It’s totally normal to have pauses in coversation - full stop. Sure, I know the feeling of quiet that may be strange, where one, or both feel hesitant or not sure of what to say. And in your example, your conversation partner clearly voiced that. But then… what? Who cares? So What?
I get that nobody wants to feel awkward. I want to be cool. I want to have cool conversations. I don’t want to be weird. Right? But where does that get me - these desires to be a certain way? I think it adds to the pressure of being a certain way, failing expecations. And then the post-event evaluation (which really hammers home the cycle of social anxiety). Ultimately it gets me nowhere. Even if something is awkward, okay! Better to learn to embrace that than to fight it. Everyone in the world who has ever had a conversation has had some sort of pause in the conversation. You could even say that everyone has had awkward pauses - if you choose to define them as that.
The remark also says more about your conversation partner than it might about the conversation. She wasn’t comfortable with the pause. Maybe she was nervous. Who knows? And if she was, fine. No big deal. Doesn’t have to mean anything significant about you.
Doesn’t matter where you look. Doesn’t matter if you choose to say nothing for a moment or two. The onus of all conversations is not on just you. I don’t mean to say we should disregard social etiquette and norms.