The question doesn't explicitly ask for "what could I have done better", although it's probably part of an answer. So I'll start there, going back to your original knowledge, to try and outline the things you could have done instead, before looking at what to do now.

Mainly because you need to understand your problem, before trying to fix it. Indeed, your problem happened because you didn't try to understand a problem (as others might see it, or at all) before trying to fix it. So this is an important mistake to avoid when you try to put it right.

The "new mums" space

You just learned there was a new mothers' space. You reacted "let's create a new dads space as well". Hmm. Do you really think parents should be divided by gender? Did you ask what led to the space for mothers being created, or how it is used, or its purpose?

So your first mistake was to jump in, rather than ask, learn, and check things first.

(As an aside this can be seen as rather "entitled" behaviour, especially as it doesn't sound like you are a new mother and you are discovering the room for the first time, yet you automatically assume you know what's best for those involved - a group you aren't part of and may have needs you don't know about.)

Linear thinking, and probing to open up the concern

A second mistake was to apply fairly linear thinking. There is a new mums room? So there should be a new dads room too! Yay for logic! But what, exactly, would have been the real benefit of a segregated room of this kind, even if not rejected by others?

A better direction would have been to inquire and discover, were any people with new babies being excluded, by having the room only for new mums, rather than a space for new parents generally or a young babies and carers space. You would have discovered whether new dads had needs, or been overlooked, and also whether anyone else with very young children would benefit from support.

You would have started to pivot towards gender and assumption-free wording, and to the function of the room rather than assumoptions about who needs a space and how they are categorised.

As it happens, you would also have learned the fact that it was used for breast feeding privacy if you had done so.

Exploring issues in the current setup

Your next mistake was to see the matter too superficially. Suppose you had asked these things, and learned it was a way to provide breastfeeding parents with privacy. You might consider if this is the same issue, or a different issue.

For example, you probe whether a new parents space might be better as it's less assumptive, and are told that actually, breastfeeders want a male-presenting free space for privacy, and that's the underlying purpose of the room, so having a "new parents" space wouldn't work.

This might spark a thought, what if a new mother wanted privacy from other new mothers (not unreasonable with work colleagues)? Should we provide a new parents space with a few drawable curtains or cubicles, or some other means, so that mums who are breastfeeding can have privacy? Do we even know how new mums think about this area and if it does what they need?

What to do now?

As you can see, equality type work is immensely valuable, but requires a different approach from much work. You can't assume you are the one with the answers. You can't assume you are the one entitled to speak first and solve problems. You need to learn listening and reflecting skills. As in science, the most valuable thing is a person who can ask good questions. You need to learn that.

It may be that this role isn't for you. If so, quietly stand down from it.

Otherwise, be up front about the error. Why? Because people will judge an excuse far harsher than honesty. "I'm new and clearly have a lot to learn. I hope to ask more and assume less, going forward, and hope you can accept my sincere apologies for jumping in without thinking through, at the last meeting."