"Now the baby is healthy and carrying on with its life, the entire family thanks everyone for their help, along with those who cared for the child. We will use all our love to protect the baby as it grows older," the grandfather was quoted as saying.
"We ask society to give our family a quiet living environment," he added.
The family come from Jinhua, in the eastern province of Zhejiang, and the paper is based in the same province.
Another report, by the state-run China News Network, cited local police as saying they had found the baby's suspected father, but would need to carry out a DNA test to confirm his paternity.
"Although the mother says she has found the father, we need to be sure... he is not sure if he is the father, and requested a DNA test," the report quoted local policeman Zhang Jianbo as saying.
"He said that if the baby was his, he would take responsibility for it," he added.
Despite early speculation that the baby was flushed down the toilet on purpose, police said that he fell into the pipe by accident, and that the mother will not face prosecution as she did not drop him on purpose.
The child, dubbed "Baby No. 59" by nurses after the number of his incubator, was released from hospital on Wednesday, hospital staff said.
The 2.3-kilogram (five-pound) baby was trapped for at least two to three hours, authorities and media reports said previously, and suffered some cuts to his face and limbs.
Chinese babies born out of wedlock are sometimes abandoned because of social and financial pressures. The country's one-child policy can also mean heavy fines for couples who have more than one baby.