Can I request to see a reference about me?

As a general rule, institutions are very reluctant to give access to letters of reference to anyone and candidates in particular. That's because they know that if they regularly shared these letters with the people being written about, letter writers would stop giving honest assessments.

As a consequence, regardless of matters of law or regulation, you should expect that universities will not give you the letter. You may have a legal way to obtain such a letter, but you will likely have to sue the university to obtain it. Whether that's worth your time or money is a case you need to evaluate for yourself -- taking into account that knowing what is in the letter does not actually change anything. It's not like the university will offer you the position you were seeking just because you know what was in the letter.

Here is a link to some information about the Family Educational Rights and Privacy Act of 1974 or FERPA for the rules and regulations in the US. In general, you have the right to see any and all records. However, there are two critical caveats with regards to LoR:

you can only access your recommendation letters after you've gotten your admission letter of acceptance and chosen to enroll in a college. If you were rejected from or chose not to go to a certain school, then you won't be able to get your hands on those letters.

Once you add a college to your Common Application, you'll see a tab show up to invite your recommender(s). Before you send those invites, you'll be prompted to read about your waiver of access and select a response.

So letters of recommendations only have to be shared once you join a specific institution and you will be asked in advance of your application if you want to waive this right. This means, in cases of rejection you have no recourse to see the letters. My guess is that this was put in place to protect the writers of the LoR. It is a really challenging situation for you to be in, since the LoR from your current advisor is surely important in landing a new position.

The other answers address the legal and regulatory aspects of the question. You should also consider the cultural aspects: Many American academics expect that applicants will choose not to read their reference letters. If you violate those expectations, they will not trust you.