An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick summary, but see Appendix A in Serre's book "Lectures on the Mordell-Weil Theorem" for more details:

Given an imaginary quadratic $K$ of class number 1, one considers an elliptic curve $E$ with CM by $O_K$, which is unique up to $\mathbb{C}$-isomorphism. Its $j$-invariant lies in $\mathbb{Z}$. Given any integer $n$, all of whose prime divisors are inert in $K$, our $E$ yields a unique integral point on $Y_{nonsplit}(n)$. That means, fixing an $n$ and letting $p$ be its largest prime divisor, any imaginary quadratic $K$ of class number 1 whose discriminant is larger than $4p$ in absolute value will furnish a unique integral point on $Y_{nonsplit}(n)$. Hence, determining the integral points on $Y_{nonsplit}(n)$ for one $n$ for which there are finitely many integral points solves the problem. Heegner used $n=24$, as did Stark.

My question is whether a similar approach has been, or can be, used in the real quadratic case;

the goal being to prove that there are infinitely many such fields of class number one.

For what it's worth, here's what would happen in my pipe dream:

1. To a real quadratic $K$ of class number 1 one attaches uniquely an abelian surface (or some other kind of object) with real multiplication by $O_K$;

2. One shows that this object gives rise to a unique rational point on some moduli space, and conversely, all such rational points would come from a $K$; ($Y_{nonsplit}$ plays this role in the imaginary case...)

3. One shows that the moduli space has infinitely many rational points.

(Since this is a well-known open problem, I'll follow the advice of the FAQ and make it community-wiki. I am a little embarrassed asking such a speculative question, but I also feel that speculation drives a lot of mathematical research.)