Heart failure is a debilitating condition that affects millions of people worldwide and has a significant impact on their quality of life and mortality rates. An objective assessment of cardiac pressures remains an important method for the diagnosis and treatment prognostication for patients with heart failure. Although cardiac catheterization is the gold standard for estimating central hemodynamic pressures, it is an invasive procedure that carries inherent risks, making it a potentially dangerous procedure for some patients. Approaches that leverage non-invasive signals - such as electrocardiogram (ECG) - have the promise to make the routine estimation of cardiac pressures feasible in both inpatient and outpatient settings. Prior models trained to estimate intracardiac pressures (e.g., mean pulmonary capillary wedge pressure (mPCWP)) in a supervised fashion have shown good discriminatory ability but have been limited to the labeled dataset from the heart failure cohort. To address this issue and build a robust representation, we apply deep metric learning (DML) and propose a novel self-supervised DML with distance-based mining that improves the performance of a model with limited labels. We use a dataset that contains over 5.4 million ECGs without concomitant central pressure labels to pre-train a self-supervised DML model which showed improved classification of elevated mPCWP compared to self-supervised contrastive baselines. Additionally, the supervised DML model that uses ECGs with access to 8,172 mPCWP labels demonstrated significantly better performance on the mPCWP regression task compared to the supervised baseline. Moreover, our data suggest that DML yields models that are performant across patient subgroups, even when some patient subgroups are under-represented in the dataset. Our code is available at https://github.com/mandiehyewon/ssldml