Scindia slammed Singh for his reported statement, that "many children die" in the month of August every year at state -run Baba Raghav Das Medical College Hospital in Gorakhpur, and termed it "insensitive".
Shastri, who was the railway minister in the Cabinet of late Prime Minister Jawaharlal Nehru, had tendered his resignation following the Ariyalur train accident in Tamil Nadu in November 1956 in which over 100 people were killed.
"By presenting the hospital's data (related to deaths of children in August in previous years), Singh has shown how insensitive he is in the matter.
"I had hoped he would follow into the footsteps of (his grandfather) Shastri and resign on moral grounds. But it is surprising Singh has not resigned yet," the Congress chief whip in the Lok Sabha Scindia told PTI.
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The former Union minister said Singh has shown his "insensitiveness" over a serious issue, which is condemnable.
Over 60 children have died at the hospital over the past one week.
On Bihar Chief Minister Nitish Kumar rejoining hands with the BJP in that state, Scindia said he has "betrayed" the mandate given to the grand alliance (mahagatbandhan of the Congress, RJD, JD-U)) and this will boomerang on him.