  The recent Fermilab measurement of the muon anomalous magnetic moment yields
about $4.2 \sigma$ deviation of $a_\mu \equiv (g-2)_\mu/2$ from the SM
prediction, when combined with the BNL E821 experiment results. In the Type-X
two Higgs doublet model with the Higgs alignment, we study the consequence of
imposing the observed muon magnetic anomaly, along with the theoretical
stabilities, electroweak oblique parameters, Higgs precision data, and direct
search bounds at the LEP and LHC. For the general study, we scan the whole
parameter space and consider two scenarios, the normal scenario with $h_{\rm
SM} = h$ and the inverted scenario with $h_{\rm SM}=H$, where $h$ ($H$) is the
light (heavy) CP-even Higgs boson. Extremely large $\tan\beta$ (above $\sim
100$) and light pseudoscalar mass $M_A$ to explain the muon $g-2$ anomaly make
it difficult to maintain the theoretical stability, causing a chain reaction of
restricting other masses as $M_A^2 \simeq M_{H^\pm}^2 \simeq m_{12}^2 \tan\beta
\approx M_{h/H}^2$. Decoupling of new scalar bosons is not possible. The direct
search bounds exclude the small $M_A$ window: crucial is the exotic Higgs decay
$h_{\rm SM} \to AA$ in the normal scenario and the LEP $e^+ e^- \to Ah$ in the
inverted scenario. After applying all the constraints, we have narrow mass
windows, $M_A \in [62,145]\,{\rm GeV}$ and $M_H \in [130,245]\,{\rm GeV}$ in
the normal scenario while $M_A \in [70,102]\,{\rm GeV}$ and $M_h \in
[100,120]\,{\rm GeV}$ in the inverted scenario. We also show that $pp \to A
h/AH$ at the LHC has a high potential to probe the whole surviving parameter
space.
