  We derive the most general sets of viable mass spectra of the exotic Higgs
bosons in the Georgi-Machacek model that are consistent with the theoretical
constraints of vacuum stability and perturbative unitarity and the experimental
constraints of electroweak precision observables, $Zb \bar b$ coupling and
Higgs boson signal strengths. Branching ratios of various cascade decay
channels of the doubly-charged Higgs boson in the ${\bf 5}$ representation, the
singly-charged Higgs boson in ${\bf 3}$, and the singlet Higgs boson are
further computed. As one of the most promising channels for discovering the
model, we study the prospects for detecting the doubly-charged Higgs boson that
is produced via the vector boson fusion process and decays into final states
containing a pair of same-sign leptons at the 14-TeV LHC and a 100-TeV future
$pp$ collider. For this purpose, we evaluate acceptance times efficiency for
signals of the doubly-charged Higgs boson with general viable mass spectra and
compare it with the standard model background estimates.
