translator is not enough, you also need to load babel. Language name can be introduced as a documentclass option and it will be applied to both babel and translator.
Note: As I don't have Swedish translator dictionary, the example uses Spanish.
\documentclass[spanish]{article}
\usepackage{babel}
\usepackage{translator}
\usepackage{pgfgantt}
\begin{document}
\begin{ganttchart}[
hgrid,
vgrid,
x unit=18mm,
time slot format=little-endian
]{7.1.2013}{13.1.2013}
\gantttitlecalendar*{7.1.2013}{13.1.2013}{
month, month=name, month=shortname, weekday,
weekday=name, weekday=shortname
}
\end{ganttchart}
\end{document}

Update:
As it seems that Swedish dictionaries doesn't officially exist for translator, you will have to provide corresponding translations in your source file:
\documentclass[Swedish]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{babel}
\usepackage{translator}
\usepackage{pgfgantt}
\uselanguage{Swedish}
\languagepath{Swedish}
\providetranslation[to=Swedish]{January}{Januari}
\providetranslation[to=Swedish]{Monday}{Måndag}
%---- Complete with missing months and days
\begin{document}
\begin{ganttchart}[
hgrid,
vgrid,
x unit=18mm,
time slot format=little-endian
]{7.1.2013}{13.1.2013}
\gantttitlecalendar*{7.1.2013}{13.1.2013}{
month, month=name, month=shortname, weekday,
weekday=name, weekday=shortname
}
\end{ganttchart}
\end{document}
