It is nigh impossible to prove that something didn’t happen, short of an explicit statement to that effect. However, there are various indications that suggest that Dumbledore did not share the prophecy with Petunia.
In Chapter Four of Half-Blood Prince Dumbledore tells Harry the following:
“There are only two people in the whole world who know the full contents of the prophecy made about you and Lord Voldemort, and they are both standing in this smelly, spidery broom shed.
Assuming that Dumbledore was not lying or mistaken (though see here) this would mean that Petunia was certainly unaware of the full contents of the prophecy.
In Chapter One of Philosopher’s Stone Dumbledore tells McGonagall the following:
“His aunt and uncle will be able to explain everything to him when he’s older. I’ve written them a letter.”
While “everything” could conceivably include the prophecy, it would seem unlikely. It would not be particularly necessary for Petunia to know about it to fulfill her role as blood protector. Moreover, Dumbledore didn’t want to tell Harry about the prophecy at all, so it would be all the more unlikely that he would want Petunia to tell him about it.
In Chapter Thirty-Seven of Order of the Phoenix Dumbledore actually tells Harry what he wrote in the letter:
“While you can still call home the place where your mother’s blood dwells, there you cannot be touched or harmed by Voldemort. He shed her blood, but it lives on in you and her sister. Her blood became your refuge. You need return there only once a year, but as long as you can still call it home, there he cannot hurt you. Your aunt knows this. I explained what I had done in the letter I left, with you, on her doorstep. She knows that allowing you houseroom may well have kept you alive for the past fifteen years.”
He says here that he explained what he had done. Nothing about the prophecy is necessary to explain what he had done, so presumably he did not include it.
In general, Dumbledore tends to operate highly secretively, revealing as little information to as few people as possible. It would seem highly out of character for him to confide something of such significance to a muggle (who might spread the information intentionally or unintentionally) when he hadn’t entrusted this information to even his closest wizard colleagues.