How to select a DSP chip for ECG/EMG signals
It sounds like you don't need a DSP chip at all.  Since this data is already measured and available on the net in digital form, you don't need a dedicated embedded system according to your requirements.

So the answer is: Any PC or laptop.  These have very good math capabilities that are probably better than most small DSPs, even though they don't specifically have DSP hardware.  You can certainly run Linux on one, the gcc compilers are available, ethernet connectivity is built in, and if it doesn't come with a SD card slot you easily add a external card reader that does.