Are there Earth rocks on Mars? Certain meteorites found on Earth have been established to come from Mars: a giant impact ejected rocks from Mars, these rocks traveled through interplanetary space, and went through the Earth's atmosphere without completely burning up. 

Might it be possible to find a meteorite on Mars that comes from Earth from the same kind of mechanism? 

Earth's gravity is stronger than Mars', so I guess you would need a bigger impact to send debris into interplanetary space. And the atmosphere is thicker on Earth, so the impacting body would need to be even bigger for enough of it to touch the ground. And you also need to give the rocks more energy so that they can go outwards in the Sun's gravitational well. So I'm guessing that impacts that might be violent enough are rarer for the Earth than they are for Mars.

Might it be possible for there to be Earth meteorites on Mars? And if so, is there any chance that we might stumble upon one?
Well if no one is going to answer this I will. The answer is we don't know for sure. We speculate that there should be earth rocks on mars but until we 'see' one and analyze it we will not know for sure.  The comments here all point to this answer.
Mars hit with thousands of Earth rocks possibly containing life following asteroid impacte talks about the Chicxulub impact and how it probably spread rocks to all the terrestrial planets in our solar system and the moons of all the planets.

The statement comes from Penn State University researchers who have calculated the approximate number of rocks from our planet large enough to possibly carry life that have made their way into space over the past few billion years.
Said the paper’s lead author Rachel Worth: “We find that rock capable of carrying life has likely transferred from both Earth and Mars to all of the terrestrial planets in the solar system and Jupiter. Any missions to search for life on Titan or the moons of Jupiter will have to consider whether biological material is of independent origin, or another branch in Earth’s family tree.”

The article cites its source as The BBC's Dinosaur asteroid 'sent life to Mars' which cites a 2013 paper published in Astrobiology Seeding Life on the Moons of the Outer Planets via Lithopanspermia:

The asteroid that wiped out the dinosaurs may have catapulted life to Mars and the moons of Jupiter, US researchers say.
They calculated how many Earth rocks big enough to shelter life were ejected by asteroids in the last 3.5bn years.
The Chicxulub impact was strong enough to fire chunks of debris all the way to Europa, they write in Astrobiology.
Thousands of potentially life-bearing rocks also made it to Mars, which may once have been habitable, they add.
&quot;We find that rock capable of carrying life has likely transferred from both Earth and Mars to all of the terrestrial planets in the solar system and Jupiter,&quot; says lead author Rachel Worth, of Penn State University.