There is probably a more elegant way to do this, but one approach would be to use the Dissolve (Data Management) geoprocessing tool. When running the tool, to make it dissolve based on spatial location, do not check off any of the checkboxes under the setting Dissolve Field(s) (optional). This will result in a feature class with one record - a multipart point. To restore the individual points, take the output and run the Multipart to Singlepart geoprocessing tool.
However, this process may not be the best depending on where you want to go with all this. A big issue that will need to be thought about is if the attributes need to be carried over from the original points. If that is the case, it may be that one or more Spatial Joins may be necessary.
Chris Donohue, GISP