Background

At my work, we have a schedule option where you get every other Friday off. It is the standard schedule all new hires are given. Today, my manager implied that I need to go into work this Friday with the phrase "If you need me I'll probably be working from home tomorrow". But she never explicitly stated "you should probably come in tomorrow". Nor did I have time to ask her to clarify the statement because she was rushing on her way out.

For context, I am working on a task that is very high priority, and she said at the beginning of the week that it needs to get done. I also had a doctor's appointment a week ago and still need to make up work time to compensate (1.5 hours exactly). The specific task requires that I be in the building, so I cannot work from home. That said, I was never given an explicit heads up that I will need to work on a Friday that I am supposed to have off; the suddenness of the statement is what concerns me and prompted me asking this question. Therefore, the ethical quandary in question is the duty to finish my task versus the right to a day off that is part of my conditions of employment.

Question

Given the situation above, am I ethically obligated to come into work on my off day?

You're not ethically obligated, especially since your boss didn't tell you to go in (extemporaneous remarks about her availability don't count). That said, it could be a wise move in terms of your future at the company.

As you progress in your career, you probably won't always have someone telling you explicitly when to work and when not to work but you will always have important deadlines and obligations that have to be met. You'll have to decide for yourself if it's worth cancelling your plans for your day off but it might be good to start thinking in terms what needs to get done (and what can wait) rather than just putting in your hours.

Why guess? Send her an email or an IM asking her to clarify whether you’re needed.

You are conflating two things here and not seeing a slightly bigger picture. The first is that you have important, urgent work to do and have taken some personal time out of the office to go to the doctor recently. From my point of view that is a good reason to go into work on a day off or work late on a few days to make sure everything is completed and signed off in a timely fashion. That way you make up the time and complete what is required.

The second thing is that you are overthinking a comment that was only meant to give you information and not imply anything. When your manager states that she is working from home if you need her for anything she isn't implying that you should be at work that day or that you could or would need her. All she is doing is saying where she will be. She probably didn't even consider whether you were working that day or not when she said it because she just wanted you to know where she is. In comments you say:

"The implication is, she will be available if I need her. Why would I need her if I'm not there? I have to be there to need her."

I don't think that that implication follows at all. I worked with someone who had a serious accident whilst on a day off (I was with him at the time as it was actually after work the evening before his day off) and our manager was working from home the next day. This meant that in spite of his not working he needed our manager to report that he wouldn't be in the following week thanks to the injury. There are countless other reasons why you might need your manager's location information even if you aren't working.

Following on from this she may not even have realized that you weren't going to be in the office at all. I had a boss who would say "see you tomorrow" when he left most Fridays. He didn't expect me to work on Saturday he just didn't always remember what day it was. My typical response was "you can come in if you like but I don't work Saturdays" and it was laughed off.

My third point is really why I'm writing this answer. You should never feel obligated to work on a day off. Ever. It actually hurts your productivity and the company in general if you don't get enough rest and sufficient down time. You need to make up any hours that you missed, and complete urgent tasks, obviously but that should not be to the detriment of your productivity. You should never feel ethically obligated to work during your time off as it is hurting rather than helping the company in most cases. You should feel obligated to take your rest and be refreshed for your work on your next shift. Some companies don't like this and expect you to work tonnes of overtime but only the most toxic ones don't understand this when it is explained to them.

Use your time off to become more productive when you are working, make sure that you complete urgent tasks on time, make up any hours that you have missed, and don't assume that because a manager gives you information about something it is because they expect you have to use it. Your manager is as human as mine who never knew what day it was!

It seems to me that your manager was just advising her reports that she was going to be out of the office but available from home. It had no expectation of anything from you. That you have this Friday off probably didn't even occur to them.

I also had a doctor's appointment a week ago and still need to make up work time to compensate (1.5 hours exactly).

If "need" means "my company forces me to" (as opposed to "I have to catch up at some point"), it means it relies on apothecary methods of measuring work time. In that case a Friday off is off, even if the world is crumbling. Because precise time management.

I am fully aware that this is not the right approach for a successful career in that company but this is not the company I would like to have a career in.

There is no ethics in business, strictly speaking, so the answer is no.

However, from the perspective of career and expectations, if you were asked to come in on a free day and something very important depended on it, then it would be the right thing to do to come in. And the right thing for the company would be to reward this dedication, as well as give you another day of your choice off.

In your situation, you were not explicitly asked to come in. The right thing to do would be to get clarification. By whatever is the best way to reach your boss, make it clear that if you are needed for the important deadline, you are ready to come in on your free day. Make your own guess at whether or not this will make a difference to the deadline, and then say either:

"By my own estimate, I believe that it would be for the best of the project if I did came. I'd like your ok on that, and I will gladly move my free day somewhen else."

or

"By my own estimate, it won't make a difference to the deadline which we will keep/miss whether or not I come in. But if you think different and request that I come, I will do so and move my free day to another date."