The final bracha of havdala must be recited with the cup of wine. The original institution of havdalah was with a cup of wine and therefore the bracha after the wine was invalid without it. However, should it be impossible to get a cup of wine, then one can bediavad rely on the bracha from shmone esrai that was said motzaei Shabbat (ata chonantanu). If one can obtain a kos, then he is required to make havdalah al hakos.
As a result, saying havdallah on motzai shabbos without wine is not allowed. That is why one who is sick and must eat on tisha b'av must first make havdallah (with a kos) before eating.
Rambam Hilchos Shabbas Chapter 29 Halacha 6
6 It is a mitzvah [instituted by] our Sages to recite kiddush over [a
cup of] wine22 and to recite havdalah over [a cup of] wine.
Although one recites havdalah in one's [evening] prayers, one is
required to recite [this blessing] over a cup [of wine].(23)
[Nevertheless,] once a person has said "[Blessed is He] who
distinguishes between the holy and the mundane," he is permitted to
perform labor even though he has not recited havdalah over a cup [of
23 In Hilchot Tefillah 2:12, the Rambam mentions the inclusion of havdalah in our evening prayers by the addition of the passage Attah
Chonantanu to the fourth blessing of the Shemoneh Esreh.
When the Men of the Great Assembly ordained the prayers and blessings
for the Jewish people, the Jewish people were needy and could not
afford wine. Therefore, these Sages ordained the recitation of
havdalah in prayer. Afterwards, when the fortunes of the people
improved, the Sages ordained that they should recite this blessing
over a cup of wine, like kiddush.
Subsequently, our people's prosperity suffered and the Sages
reinstituted the blessing into the prayer service. Nevertheless, in
order to prevent the matter from being totally dependent on
socio-economic factors, they established the following guidelines: A
person should recite the havdalah prayers in the Shemoneh Esreh of the
evening service. Afterwards, he should try to obtain wine in order to
fulfill the mitzvah as the Sages prescribed. If, however, this is not
possible, his recitation of the prayer in the evening service is
sufficient to fulfill his obligation.