The UK offers a Post Graduate Certificate of Education which certifies teachers domestically and is useful in the international school system. As an American teaching abroad, I am looking for an equivalent. What is the US equivalent to this?
There is no such certificate.
To teach at a public school in the US, you will need to be licensed by the state you wish to teach in. If you have a BA/MA in education from an accredited university, it is often trivial to get this -- you normally have to pay a fee, get fingerprinted, and provide your transcripts. But some states are worse than others -- you may have to take exams to prove you're qualified. I've heard stories about having to redo your BA/MA at an in-state school, but I have not been able to confirm such reports. There is also such a thing as "emergency licensure" -- for example, if a school can't fill a key post, they can hire someone temporarily that normally wouldn't be eligible (e.g., a physicist without teacher training to be a physics teacher).
At a private school in the US, each school has their own rules. State licensure and/or a degree in teaching is a good first step, but each school sets their own rules.
To teach in an American school abroad, things are even less well-defined. To my knowledge, "American Schools abroad" do not have any oversight by the US Department of Education, so there are no uniform standards. Each school sets their own rules in accordance with local laws. Fluency in English and some previous teaching experience is often enough, though in some saturated markets (e.g., Geneva), and in some countries with stricter local policies, the requirements can be much higher.
The role of a postgraduate degree in education often fills the need of the PGCE. In the UK and in other Commonwealth countries, you get your degree first, like a physics degree for instance and then do your teacher training.
The US simply does not use that model, most of the training in regards to your ability to teach is done at the postgraduate level. There is, of course, no uniformity in how the US approaches this, each state has it's own licensing requirements