This seems like a case of getting the tone badly wrong and then someone responding badly to that. It seems a bit odd, to me, to say in an email asking to meet up with a brand new mum, you write about wanting 'me time' and being 'stuck in with your DC' - especially if, as you say, the kids were invited too. I would have assumed from your OP that you expected the kids would not be there - yours at any rate.
She is clearly not going to be at that point yet and may have felt got at, and that she is being dull and pathetic by wanting to stay in with her baby. Maybe she is even scared of how she will manage when she leaves the house (I know it daunted me) so to save face, and being a bit nettled by your apparently unsympathetic remarks, she sent a defiant reply that has in turn got your back up.
I'm sure you didn't mean it that way; it sounds like you were excited to think that you wouldn't be so isolated anymore and wrote things that she then misread. As mistletoekisses says, email is bad for these things.
Of course, we don't know the exact wording of both your email and hers - and I am not asking you to post them here either. But it might be worth a think about whether this is a communication that's gone unnecessarily off track. As has been said, it would seem odd to cut off a friend for one snippy remaark, which suggests that this has its roots in existing resentment. I know you have said that she didn't understand the situation when you first has DC, but then as you'll know, that's pretty common with childless friends, so one worth letting go, if you can.
You have a chance to be the bigger person here. I would ring, or send a nice card., saying you understand she is totally wrapped up in her DC at the moment, you'd love to come and see them both and will make the tea when you get there , and that if she fancies a trip out of the house later on with kids in tow and wants some company, she only has to ask you. Then leave it a bit and see what happens. If you really can't bring yourself to do that, I would keep quiet for now rather than make a big gesture of cutting off the friendship that you may regret later - if, that is, she is really a friend.