John Tobin (50). of McClurg Court, had been drinking with a 27-year-old woman before the 18-month-old went missing in May last year.
The woman told a court she left the child in the care of Tobin before leaving to meet a friend, but he claimed she had taken him with her.
Members of the public raised the alarm when the child was spotted alone in Albion Street at 8.40pm on May 29, a short distance from Tobin’s flat.
An investigation by police about who was responsible for the child led police to break down Tobin’s door where they found him they found him asleep in bed drunk.
After being woken up Tobin began shouting and swearing at the officers.
Tobin was cleared of the neglect charge at Hamilton Sheriff Court after Sheriff Allan McKay said he had doubts about the evidence given by the woman.
But the sheriff did convict him of a breach of the peace for abusing police officers and fined him £300.
Sheriff McKay said: “If I’m to find beyond reasonable doubt that you had care of the boy at that time then I have to believe the woman and I have some difficulty in believing her and I am unwilling to trust her evidence.
“I am however in no doubt that you reacted in an appalling way with these police officers and will find you guilty on the second charge you faced, but will deal with that by monetary penalty.”
The woman was also found to be in contempt of court after failing to appear at two previous trial diets and will be sentenced next month.