arcgis desktop - Pacific-wide Projection for distance?

I have a data set of ~3500 points located across the Pacific Ocean, and am looking to measure the distance from each point to it's nearest coastline to identify the "host" country. I had originally used Plate Caree, but now believe that it doesn't maintain distances accurately? I am hoping not to need a country by country approach, but could someone confirm Plate Caree does / does not maintain accurate (reasonably accurate) distance, or suggest another? Also, I am fully a user of ArcMap, not QGIS, etc. so 'how to' answers outside ESRI aren't helpful in this instance.