Much of Piaget's work has now been discredited for the simple reason that it did not accurately follow the scientific method and the thorough approach that modern science has come to represent. In fairness to Piaget, he worked in a different era with different values and methods that perhaps were not as stringent as those used in later decades. In addition, at heart Piaget was a theorist working on his own method rather than trying to prove any preexisting material; work dealing with childhood development was still in a nascent state. Regardless, he's been criticized for using small control groups and making generalized assumptions about children based on these small samples. Observing his own children was the genesis of much of his theory.
The field of sociology didn't exist when Piaget was doing his work with childhood development, so it's natural that he would not have known the core assumptions in the field. Piaget often placed importance in biological factors rather than social ones when researching the development of a child. Later researchers recognize that both genes and social influences are factors in child development.
To Piaget, coveting material objects was perfectly acceptable and was even considered a sign of progress. Today, researchers recognize the cultural bias in this sort of approach. While most Western audiences would be comfortable with a child wanting or owning a toy, other cultures may see this as a sign of weakness or materialism. Cultural awareness was not Piaget's strong suit.
Piaget's theories are often broken down into stages that try to succinctly describe entire developmental periods of childhood in basic terms. In the 21st century, researchers recognize that children often develop at different speeds and in different ways. In Piaget's "The Psychology of the Child," he explains that the first stage is called the "sensorimotor" stage because "the infant lacks the symbolic function...he does not have representations by which he can evoke persons or objects in their absence," but Piaget does not have any basis for saying when children will develop the symbolic function nor any awareness of what the newborn is experiencing.