In brief, no, this is not permissible. Two issues:
1 - See here regarding the problem of bleeding and causing a wound on Shabbat. Some say it is a prohibition of shochet - slaughtering. Others say this is a prohibition of dosh b/c blood is "extracted" from blood vessels and capillaries.
The article does specifically mention giving a blood test on Shabbat, which is "more sever" than donating blood to someone else. Even that is prohibited unless this is a case of pikuach nefesh (which I shall define, later).
You mentioned Rosh Hashanah, which is not Shabbat but a Yom Tov. The only melcahot permitted on Yom Tov are those needed for food-related preparation such as slaughtering an animal that you intend to eat on Yom Tov, itself. Donating blood is not related to eating, so it would be prohibited on Yom Tov.
2 - Pikuach Nefesh has a definition that applies to saving a specific life of someone who's life is in imminent danger. This implies two things - A) It is the life of a specific person (i.e. - not a random person whom you don't know about, and b) if you don't immediately act, that person may die.
Regarding a blood donation, you don't know about anyone whose life is in imminent danger NOW. This is a remote possibility. See this article near the middle regarding administering a smallpox vaccination (which could be more severe than a blood donation.) Citing:
Rav Shlomo Zalman Auerbach zt”l discusses the situation where a
particular illness is theoretically life-threatening, but the chances
of it being so are relatively slim. Does this also warrant chillul
He maintains that if most people would act immediately due to the
urgency of the matter, then this is a case of safek pikuach nefesh and
chillul Shabbos is required. However, if most people are unconcern
about such an illness, chillul Shabbos is not allowed as there is no
sakanah. The example that he gives is the smallpox vaccination.
Although technically speaking, once the doctor says that the child
needs the vaccination, one should administer it as soon as possible,
nevertheless, since most people did not do so, as the need is not
urgent, one is not allowed to transgress the Shabbos for it (Shu”t
Minchas Shlomo, vol. II, #29.4 [in later editions: vol. II-III,
With a blood donation, it is quite obvious that "most people" are "unconcerned of an illness". As a matter of fact, you don't even know IF there is ANY kind of "illness" as you don't even know where your blood is going.