The answer appears to be a qualified 'Yes.' As reported by VentureCake (discovered via Slashdot), VMware's ESX appears to be derived from Linux in a material way, and has been notified of such over a year ago (and repeatedly since then). Yet it has not disabused the accuser (Christopher Helwig, the Linux SCSI storage maintainer and one of the top 10 contributors to the Linux kernel) of the notion.
If true, VMware has a problem on its hands. But the problem is easily solved by simply abiding by the GPL, and may not involve giving away the crown jewels, as it were. What seems to be at issue is a driver that ESX requires:
The only way to load vmkernel is by vmkmod, a driver that requires Linux. Take away Linux and there's no way to load vmkmod and start vmkernel.
It's possible to ditch, remove, or crash Linux after vmkernel has virtualized it - but you wouldn't be able to get to that stage without Linux being used to load vmkernel.
[Helwig writes:] "VMware uses a badly hacked 2.4 kernel with a big binary blob hooked into it, giving a derived work of the Linux kernel that's not legally redistributable. I unfortunately don't have enough copyrights on that particular version to sue them. I do object to use of any open-iscsi code of my origin to be used with it, though."
I personally think VMware does more good than harm to open source, and would prefer to see this resolved in a way that leaves VMware (up 79% on its first day of trading publicly) and open source well off. To get to that point, VMware has some legitimate questions to answer. Silence here isn't golden.