By default, the insert will be rolled back if the "after insert" trigger encounters an error. But, you can write the "after insert" trigger to handle the exception locally and not cause the triggering insert to be rolled back. Your trigger just needs to look something like this:
create or replace trigger [your_trigger_name]
after insert on [your_table_name]
for each row
begin
[your_procedure_name (parameters, etc.)];
exception
when others then
[null; or whatever you want to happen if an error is encountered];
end;
/
