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Global ocean circulation has a significant impact on climate variability, where ~80% of the ocean energy transfer occurs in small-scale processes. While the existing record of altimetry goes back thirty years and has enabled the assimilation of gridded sea surface height maps, their operational resolution of 0.25° is not high enough to study these mesoscale eddies, and we are therefore in need of methods that can improve their resolution globally. The recently launched SWOT satellite with ~2km resolution now offers the first data record with sufficient resolution to reveal these processes in observations, and offers the possibility of drastically improving sea surface state maps. However, its sparse temporal and spatial record brings challenges for global assimilation. We propose a generative machine learning approach to downscale existing gridded Level 4 sea surface height to the fine resolution of SWOT. Our methodology involves two steps: first, training a conditional diffusion downscaling model on high resolution simulated data as a prior joint distribution over sea state observations, including height, temperature and salinity. Secondly, a data assimilation scheme via a Bayesian posterior formulation that generates high resolution sea surface state maps assimilated with a set of observations. We evaluate our methodology both in simulated and observing system experiments that demonstrate the efficacy of our approach as well as their scalability to global context in evaluations of major currents. Under the Bayesian formulation we also find that the diffusion model produces well calibrated predictive uncertainty estimates, which further underlines the applicability of diffusion models as a computationally efficient method in this domain. Our high resolution sea surface height maps open up new insights into mesoscale eddies.