if you’re asking for something you can connect directly with ArcGIS to get access to observations as points and data tied to the points, the answer is no, as far as I’m aware.
you can download observations from iNaturalist as, say, a CSV file and then load those into ArcGIS, but that is not real-time, since you’d have to generate the CSV (ex. https://www.inaturalist.org/observations/export?projects[]=baltimore-county-public-schools-bioblitz)
and then load the data into ArcGIS each time you want to get new data.
the old iNaturalist API (https://www.inaturalist.org/pages/api+reference) does provide a way to generate CSVs more dynamically, but it has a limited number of observations it can return per response, and you’d still have to manually load the data into ArcGIS each time anyways, i think. so neither of these options would effectively provide real-time data.
as kueda mentioned in his earlier post, you can get close-to-real-time raster tiles via the API, and you can pull these into ArcGIS fairly easily, as described here: https://forum.inaturalist.org/t/looking-for-inaturalist-observation-map-visualisation-suggestions/7322/10.
the raster tiles themselves don’t actually have data associated with the markers, since they’re just image files, but there is a companion set of UTFGrid tiles (http://api.inaturalist.org/v1/docs/#/UTFGrid) that you could sort of get information from, although you can’t really easily connect these to ArcGIS without a lot of coding. you can use the UTFGrid to return details for the latest observation at a given marker shown on a corresponding raster tile, or if you’re a little more adventurous, you could try to create some custom observation density maps, as described here: https://forum.inaturalist.org/t/looking-for-inaturalist-observation-map-visualisation-suggestions/7322/11, although generally it would be better nowadays to use iNaturalist’s new grid tiles (http://api.inaturalist.org/v1/docs/#!/Observation_Tiles/get_grid_zoom_x_y_png) for this kind of map.
and that brings us to the last mapping option, which is just to use the maps within iNaturalist itself. the tradeoff there is that you can’t use your own custom layers. so you’re limited to the basemaps, etc. that iNaturalist provides.
finally, if you’re thinking of converting data from Esri over to iNaturalist, then this may be of interest to you: https://www.inaturalist.org/observations/import#csv_import.