The objective of a 1:1 is to allow an open conversation between manager and the team member

Yes, but that doesn't extend to being able to allow a (mutually agreed upon) third party to attend these meetings. For example, it is perfectly fine to want to invite a third party to an HR meeting when you e.g. feel under the gun.

However, context applies. If this is about a yearly performance review, I wouldn't particularly raise any issues if she invites a third party. As long as it's her free choice, it shouldn't be a problem, unless you suspect this person may have forced her to invite her.

Then again, if this is about a daily working schedule and her refusal to work 1-1 with any male in the workplace, this can become an obstacle to her performing her actual work duties; at which point it's no longer a request that is easily accommodated.

to me, a private environment such as a conference room promotes that objective, as well as allowing for employee privacy.

By inviting the third party, she gives up her right to privacy regarding the topic of the meeting, at least in regards to the third party she invited.

Having 1:1 in such an environment unhindered from public distraction is a benefit and something to be cherished.

Public distraction can be a valid argument, in the right context. Not wanting to do the meeting in a loud restaurant, for example, is perfectly reasonable.

"something to be cherished" is your opinion, and clearly not hers since she actively suggests meeting in a public place. Don't force your opinion on others, and don't build your argument in a way that implies that your personal opinion is universally and objectively correct.

Because if you allow your opinion to be stated as the way things will be done, then what is stopping her from not budging on her personal opinion on no private male-female one-to-one contact?
The only way you can win that argument is by pulling rank, which negates your entire "belief in collegiality in the workplace and minimization of hierarchy between team members" position.

It's reasonable to ask her to reasonably compromise from her stance, but it's only fair for you to do the same then as well. Barring any obstacles to the actual work needing to be performed, there is no reason that the two of you cannot work out something that works for both parties.

As a manager, I believe in collegiality in the workplace, minimization of hierarchy between team members, and open communication. I value growth in my team members and see myself as a guide / mentor, not someone who commands and dictates from above as a boss to be obeyed without question.

The corollary here is that you shouldn't then enforce the 1-1 without question. If you were to force the issue of it being only you and the team member, then you are in fact dictating how the meeting should proceed.

Excessive focus on gender roles and altering behavior solely because a colleague is a woman goes against such ideals and company culture at my employer.

You're overstepping your boundary here. This should be treated no different from a male team member asking for a support person to attend this meeting, for whatever reason (as long as it is of the team member's own volition).

It's perfectly okay to raise this question in regards to any work-related obstacles this third-party-requirement would pose. It is not okay to raise the issue based solely on the specific context of why this person is making that request.

Treat the request as objectively as you can, and therefore don't build your argument based on her specific reason for asking something which is completely reasonable (in the right context).

To put it shortly, you're really close to enforcing your personal ideology on your team.
Regardless of what your ideology is, this is not okay. You have some freedom as to how you choose to manage your team, but promoting inclusivity also means respecting reasonable boundaries of your team members.

If anything, she didn't even need to justify her request for a third party to be present. Having explained her reasoning is a form of open communication. If you value open communication, you just received it and are now trying to use that against her request, which is exactly how you get your team members to no longer engage in open communication with you.

She also mentioned that hierarchy is more valued in the work culture of her country.

This is a flaw in her argument, in my opinion.

As much as everyone should always respect other people's cultures and customs, she's not working in her country or their work culture. It's perfectly understandable to show some acknowledgement that she's used to a different work culture and to account for that when judging her actions, but that is not the same as you now being forced to follow another culture's work standards because she would prefer it.

I talked to her to find out why and she responded that interactions between men and women are not usually as open as the United States, and are somewhat viewed with caution.

There's a reasonable line to be found here. Accommodate what she feels comfortable with in so far as it doesn't obstruct the work activities, but don't let the work activities be unreasonably obstructed by it. I say unreasonably obstructed, because it's very easy to (intentionally or not) misconstrue the smallest of accommodations as an obstruction.

As she is a Muslim, let's take the example of daily prayer. When scheduling a long meeting, it's perfectly reasonable for her to ask you to account for her being able to take a break for prayer. It would not be reasonable to outright refuse any long meeting even when breaks are accommodated.

To summarize

• Your opinion on whether men and women can engage in a private 1-1 is of little to no importance here.
• At face value, her request for a third party is not unheard of, regardless of her reasoning. She shouldn't even be required to elaborate her reasoning here, but she chose to openly communicate this with you.
• Assuming the request does not reasonably hinder the expected work activities, there is no issue with trying to find a workable compromise for the both of you. This entails both parties giving in to some degree.
• Assuming the request does reasonably hinder the expected work activities, this is a point that should be raised, purely from a work perspective.
  • Build your argument on the impact it has on the work activities.
  • Don't build your argument based on your disagreement with her personal opinions and feelings.
• Collegial behavior includes mutual respect for each other's opinions and what the other person feels comfortable with. If you pride yourself on being a collegial manager and not one who pulls rank, then manage your team accordingly.

The context of the question here has actually muddied the water in terms of the workplace interaction that is taking place. When it comes down to it, this is no more than an employee asking for a third party to be present during a meeting.

If you cannot express why this is a bad idea based solely on workplace considerations, and without basing yourself on (a) disagreeing with her personal opinion or (b) refusing to adapt your management style to your team member, then you don't really have a valid reason to refuse this.

You could force (b) by pulling rank on her, but then you're no longer managing your team in the way you just presented yourself. This is your decision to make.