No, Earth is way more massive than Mars and won't lose its atmosphere as easily. The upper layer of the atmosphere is ionized by solar radiation. Instead of simply stripping our atmosphere away, the solar wind would induce a magnetic field in this upper ionosphere. And this induced magnetic field would exactly cancel the magnetic field from the solar wind, ultimately deflecting the solar wind around Earth. This is exactly what happened on Venus and is one of the reasons why Venus still has an atmosphere.

The same thing also happens on Mars, but because Mars is significantly less massive it still lost most of its atmosphere over time from various processes.

But you are correct with your idea that Earth would lose its intrinsic magnetic field if the core cooled down and solidified.