Silence is Golden by Rabbi Shmuel Silber According to some he was so shocked that he was emotionally paralyzed. He couldn't react, he couldn't even cry. According to others his faith was so solid and unshakeable he forced himself to continue with his duties. We see from his burning the Rosh Chodesh Musaf that he did indeed react.
R. Berel Wein contrasts the reactions of Aharon HaKohen and Iyov in dealing with personal tragedies
Aharon’s reaction to this is silent acceptance of the realities that now face him. He does not rail against perceived injustice, as does Iyov. Nor does he withdraw from the fray of life and go into seclusion, as did many others when faced with similar tragic situations. Aharon becomes the paradigm for how humans are to deal with the “eighth day” – with life and its ups and downs.
Resilience and silent inner strength engendered by faith and acceptance of God’s will are the weapons of living on in spite of all that the “eighth day” imposes upon one’s life. These words are much easier to write and to read than to actually implement. Yet the Torah expects no less from us than it did from Aharon. Life and our contributions and meaningful behavior towards making it better and stronger are always played out on the background of the “eighth day.”
The Fire of Nadav and Avihu
As for Aharon's silence, it is most fitting of all. Who realized more
than the aged priest the great dangers inherent in unbridled religious
enthusiasm? Who understood more than he did, as he stood powerless to
stem the crowd excitedly clamoring for a golden calf, that the best
intentioned acts of worship can so easily degrade into coarse and
corrosive idolatry? His beloved children, whom he hoped would one day
succeed him in the service of God, now lay dead at his feet, but how
could he blame the Deity for their demise? Nadav and Avihu were
consumed by an inner fire that could not but be matched by an outer
fire from above. Did the poet king perhaps have them in mind when he
bid his beloved to
Place me as a signet upon your heart and as a seal upon your hand, for
love is as powerful as death and yearning as cruel as the grave, its
flames are the flames of God's consuming fire! (Shir ha-Shirim 8:6).
And did God Himself perhaps have them in mind as well when He twice
warned Moshe at the revelation at Sinai to distance the people and the
priests from the mount, lest "they break forth to ascend towards God,
and He then break forth upon them" (Shemot 19:24)?
All religions speak of Divine mercy and love, inviting us to lovingly
embrace God in turn. And who is there that does not want to experience
that connectedness and concern of God in their own life? Who is there
that does not want to serve God sincerely with enthusiasm and passion?
One of the most unique contributions of the Torah to religious
discourse, however, is that worship of God, if it is to be genuine,
must also be predicated upon reverence and fear, for reverence
provides the restraints that love refuses to recognize. Nadav and
Avihu tragically perished at the very moment that God's presence was
most intensely felt. But through their death, the two brothers
communicated a fundamental lesson that was to inform the service of
the Mishkan and Temple (and for that matter the synagogue) for all
time. "Moshe said to Aharon: this is in accordance with what God
spoke, for He said that 'I will be sanctified by those that are close
to Me, and I will be glorified in the sight of all of the people'."
Too Close for Comfort by Hillel Liss
Now we can see why the Torah must comment on the fact that Aharon is silent. Aharon has learned the lessons which Hashem had just taught and accepted Hashem's will. The Torah develops this idea of maintaining a distance from Hashem in the rest of Parshat Shemini by contrasting purity and impurity in the later Pesukim. This is also why Hashem says "And separate between Kodesh and Chol, and between impure and pure."
Ramban: Aharon had been weeping, but on hearing Moshe's consolation (Shmini 10:3) he stopped.
Seforno: He found comfort in the knowledge that Hashem's Name had been sanctified by his sons.
Rashi: As a reward for his silent acceptance of Hashem's decree, Aharon was honored by having the following mitzvos (10:8-11) addressed to him.
Rav Shimshon Refael hirsch points out that this is an indication that the greater the people, the more strict Hashem is with them. As a result, Aharon could not respond because he realized that Hashem must answer the slightest deviation of his great ones. Rav Hirsch also points out that this is one of the many things told to Moshe that were not written down and, if not for this incident, would have remained in Torah Sheb'al Peh.