Rambam, Hil. Retzichah 1:2-3 writes:
מִצְוָה בְּיַד גּוֹאֵל הַדָּם [לַהֲרֹג אֶת הָרוֹצֵחַ] שֶׁנֶּאֱמַר (במדבר לה יט) "גֹּאֵל הַדָּם הוּא יָמִית אֶת הָרֹצֵחַ". וְכָל הָרָאוּי לִירֻשָּׁה הוּא גּוֹאֵל הַדָּם. לֹא רָצָה גּוֹאֵל הַדָּם אוֹ שֶׁלֹּא הָיָה יָכוֹל לַהֲמִיתוֹ אוֹ שֶׁאֵין לוֹ גּוֹאֵל דָּם בֵּית דִּין מְמִיתִין אֶת הָרוֹצֵחַ בְּסַיִף: הָאָב שֶׁהָרַג אֶת בְּנוֹ. אִם הָיָה בֶּן לַנֶּהֱרָג הֲרֵי זֶה הוֹרֵג אֲבִי אָבִיו מִפְּנֵי שֶׁהוּא גּוֹאֵל. וְאִם אֵין לוֹ בֵּן אֵין אֶחָד מִן הָאַחִין נַעֲשֶׂה גּוֹאֵל הַדָּם לְהָמִית אֶת אָבִיו אֶלָּא בֵּית דִּין מְמִיתִין אוֹתוֹ.
There is a Mitzvah in the hand of the redeemer of blood to kill the murderer, as it says, "The redeemer of blood - he will kill the murderer." Whoever is fitting to inherit, he is the redeemer of blood. If the redeemer of blood doesn't want to, or is unable to, kill him, or he does not have a redeemer of blood, the courts kill the murderer by the sword. If a father killed his son: If the victim has a son, he kills his grandfather, because he is the redeemer. If he does not have a son, none of the brothers are made a redeemer of the blood to kill his father, but rather the courts kill him.
Important parts for our question:
- The heir is the redeemer of blood.
- If there is no redeemer of blood, the courts kill the murderer.
- If a man kills his son, and there are no grandchildren, the courts kill the murderer.
According to the Rambam himself (Hil. Nachalos 1:2), if a man dies with no children, his father inherits him. If the heir is the redeemer of blood, why isn't the father given the opportunity to redeem his son's blood? It would seem, then, that the murderer cannot kill himself to avenge his son's blood.
Another proof (which also fleshes out the logic behind this halacha) can be brought from the Rokeach to Rambam, Hil. Rotzeach 1:3:
אין אחד וכו' להמית את אביו וכו'. לאו דוקא מיתה, דהוא הדין הכאה או קללה או ביזוי אף שהוא ראוי לכך, כמו שכתב רבינו סוף פרק ה' דהלכות ממרים עיי"ש.
"None of the [brothers are made a redeemer of the blood] to kill his father." Not specifically killing, as this is the law for hitting, cursing, or embarrassing, even though he is fitting for this, as our teacher wrote at the end of Mamrim, chapter 5; see there.
Well, let's see what he has to say there. Rokeach seems to be referring to Hil. Mamrim, 5:15, where he makes direct reference to our halacha that the brothers cannot be redeemers of blood, and in halachos 12-15 there he discusses several similar halachos. All of them boil down to: since hitting, cursing, and embarrassing one's parents is a Torah prohibition, even if the parent deserves it, the son should not be the one to mete it out.
The same thing would seem to apply here: even if, theoretically, the father is eligible for being a redeemer of blood, there's still a prohibition involved in the father killing himself and so it should be carried out by someone else if possible.