No, this is not possible... HP Smart Array controllers and LSI have completely different on-disk RAID metadata formats.
This will not work.
The cable connecting the Smart Array P600 to the Storageworks MSA60 enclosure is not eSATA either. It's going to be a SFF-8470->SFF-8088 4-lane SAS cable.
So, what problem are you trying to solve? You have a new server. Obviously it's not an HP server, so the expectation that you'd be able to just reattach the external storage is wrong. However, you can use a modern HP Smart Array controller in your new server to connect to the external storage. HP Smart Array P800, P411, P212 or P812 RAID controllers would all work for this purpose.
But since you're also working with a new server, do you have an option to use the new server's resources to allow you to copy the data from the old setup to the new? HP MSA60 storage enclosures are very much outdated now. I'd expect that your new server hardware can accommodate larger disks or a more appropriate external storage setup. You only seem to need 6TB of usable space, so it would definitely be safer to copy data rather than risk compatibility issues in retaining your old gear.
For a ProLiant DL380 G6 specifically, you'll want to go with a Smart Array P812 controller or a P212 with additional cache and battery. Both can be found used/cheap. Firmware updates are going to be very important to ensure everything is updated to the right revisions. Download the HP Service Pack for ProLiant to bring firmware to the proper revision levels after you've connected everything. Take backups before you try this, if possible.