It’s very difficult to watch a loved one deal with a terrible loss. And while the friend acted in good faith, it appears she might have overdone her outreach. It's a positive thing that the friends have a close enough relationship that the bereaved felt comfortable in communicating how she felt.
I don't think the friend needs to apologize. What she might do is tell her bereaved friend that she hasn’t experienced this kind of loss and she doesn't know the best way to be supportive. She might ask the bereaved friend to let her know what she can do that will be helpful and make her feel comforted. And then she should take her cues from the bereaved. If the bereaved can't communicate something tangible to her, the friend might keep in touch by email periodically, maybe once a week at first. The bereaved friend needs some time to process the loss; this is something she has to work out by herself. But it’s important to keep in touch, allowing her the time to fully grieve.