You need to put order of which ROW_NUMBER will generate numbers. Existing ID is good option here.
declare @maxID int;
set @maxID = (select MAX("ID") from "table");
insert into "table"
select @maxID + ROW_NUMBER OVER (ORDER BY "ID"), 'newData', "col3", "col4"
from "table" where "col2" = 'oldData';