Triangulate with Ping
The answer is YES, but with a lot of caveats.  Let me re-phrase the question into a more useful form:  

Using only Ethernet packets, on an Ethernet network, can you determine the physical cable length from one device to another? 

There is a standard called the Precision Time Protocol (PTP), which is used to synchronize clocks of devices on a network.   The basic protocol can achieve sub-microsecond accuracy, but there are ways to get accuracy down to 10's of nanoseconds.  

Part of this protocol requires hardware-based timestamping of Ethernet packets in order to measure and calculate the "time of flight" of a packet across an Ethernet cable.  This time of flight measurement is then used to adjust for any clock skew across the network.  In our case, we only care about the time of flight.

While this does work across Ethernet switches and routers, to get an accurate time of flight the switch needs to support PTP and be involved in the measurements.  PTP does support the involvement of the switches and routers.  Because PTP uses Multicast packets, it won't work over the Internet.

More and more Ethernet controllers support PTP (even some PC motherboards support it), although switch and router support is lagging by quite a bit.  

In theory, PTP can do this.  In practice, I don't know if you can extract the time of flight data from this clock syncing protocol-- but the data is there.  It might require some somewhat custom implementations of PTP, and definately requires switches/routers that are rare or not quite on the market yet.  I am also massively generalizing PTP.  I know the hardware side of PTP, but there is a lot of software that I don't know completely.

Another alternative to PTP is to use Ethernet Phy's that can measure the length of the cable.  It turns out that many Gigabit Ethernet Phy's can do this now, but those features are rarely exposed past the device driver level.  The Phy's do this by using time-domain reflectometry, and can also measure the distance to the break in a faulty cable.  

The main problem with any of this is that it measures cable length, not physical position.  If the cables were measured accurately, and they were stretched in a straight line, then that could determine physical location.  But that never happens.  It also requires that the network infrastructure is under your control, which it is probably not.  

Of course, what I describe is nothing like using Ping times across the internet to get a physical location.  I agree with others that this is not going to be very useful, and probably won't get you anything more accurate than what Continent you're on (if that).