No, it is not possible to integrate Google Street View in ArcMap as the functionality is currently not supported.
However, ArcMap layers can be converted into Keyhole Markup Language (KML/KMZ) files, and imported into Google Earth. Data from Google Earth or Google Maps can also be loaded into ArcMap via the conversion of KML/KMZ files into layers. However, the data from Google Earth or Google Map can only be loaded into ArcMap if the KML/KMZ file is version 2.0 or older. Refer to the links below on how to convert layers in KML/KMZ files and vice versa.
ArcGIS Help: KML To Layer (Conversion)
ArcGIS Help: Layer To KML (Conversion)
There is a third-party widget, called the Google Street View Widget, that creates a web application that integrates Google Street View with ArcGIS for Desktop via ArcGIS Viewer for Flex. However, since this widget belongs to a third party, Esri does not support its use.