He was not only feeling guilty about ordering the death of his own brother. His guilt also consisted of ordering the death of his "father's son" which translates into Michael feeling that he BETRAYED his father's trust and his mother's love by having his brother, their other son, murdered. Let's keep in mind that by Vito handing the reigns over to Michael, he trusted that Michael would take care of and protect the family in the same manner that Vito had protected the family.

So Michael felt guilty after finally realizing that he betrayed his father because he did not do what his father had trusted him to do in protecting his family.