Absolutely not. He lied when he told Eve "You positively will not die". (Gen. 3:4)
That was a lie. Eve and Adam did die. God told them the truth. Satan misled Eve.
Further, Satan slandered God, implying that by eating the fruit, they could be like God, free to make their own self-determination to decide what is right and wrong, good or bad,completely independent of their Creator. That too, was a lie, one still believed by millions today. History and current events prove conclusively that man cannot successfully make correct moral choices without direction from God.
Satan has since slandered humans, as he did Job, asserting that no one will serve God for unselfish reasons. Revelation 12:8 says that Satan is "misleading the entire inhabited earth", still spreading lies and false teachings in opposition to God.
Jesus did not overstate matters when he called Satan the father or originator of the lie.
Humans can indeed determine what is right or wrong, and gods don't play into it.
In addition, as far as the narrative goes, Adam and Eve died because god barred them from the garden, not directly due to eating the fruit.
Genesis 3:
21 The LORD God made garments of skin for Adam and his wife and clothed them. 22 And the LORD God said,
The man has now become like one of us, knowing good and evil. He must not be allowed to reach out his hand and take also from the tree of life and eat, and live forever. 23 So the LORD God banished him from the Garden of Eden to work the ground from which he had been taken. 24 After he drove the man out, he placed on the east side[
e] of the Garden of Eden cherubim and a flaming sword flashing back and forth to guard the way to the tree of life.
The serpent told them that when they ate the fruit, they would be like God and know good from evil, which God later acknowledged in that above passage. The serpent also said that they would not certainly die, which was true, because eating the fruit did not kill them. They died because God acknowledged that if they continue eating from the tree of life, they'll live forever, and so he barred them from eating from the tree.
The problem lies with god, and not the serpent. The serpent told the truth. Besides, most Jews I've talked to don't even believe that the serpent was Satan. They typically don't even have a central antagonist figure like Satan. These connections were all formed later by Christians.