You can use your ICAO licence to get an FAA private certificate under FAR 61.75. Think of it as a validation of your ICAO licence to allow you to exercise the privileges your foreign licence for private-only operations flying US registered aircraft.
You will be limited to the more restrictive limitation(s) of your foreign licence *and* the FAA PPL. If your foreign licence doesn't allow you to fly an aircraft with certain features or in a particular operation eg night, then the 61.75 will be similarly limited. If your foreign licence allows certain privileges but a US Private Certficate does not, then you will be restricted to those operations permitted by the US Private Certificate.