You can use the Path Distance Allocation tool to get the nearest source to each cell on your raster. Let's call the result of using the tool the source_raster. The source here would be the unique ID of your health facility points. This will be your service area raster if you just want to know the nearest health facility.
If you want to distinguish areas that are unserved because it takes too long to get to them from their nearest health facility you would then use Raster Calculator to trim your time_raster to your desired maximum time (max_time). An example syntax for Raster Calculator:
[service_areas] = Con([time_raster] > [max_time], -1, [source_raster])
This will give you a categorized service area raster, with the value being either -1 for unserved or the unique ID of the health facility.
You could then use Raster to Polygon to convert the raster service areas to polygons that cover the entire area of the raster.
Note that when you name your rasters I think you are limited to 13 characters, so use short but descriptive names.