CERN-LHC. The invariant differential cross section of inclusive $\omega(782)$ meson production at midrapidity ($|y|<0.5$) in pp collisions at  $\sqrt{s}=7$ TeV  was measured with the ALICE detector at the LHC over a transverse momentum range of $2 < p_{\text{T}} < 17 \text{GeV}/c$. The $\omega$ meson was reconstructed via its $\omega\rightarrow\pi^+\pi^-\pi^0$ decay channel. The measured $\omega$ production cross section is compared to various calculations: PYTHIA 8.2 Monash 2013 describes the data, while PYTHIA 8.2 Tune 4C overestimates the data by about 50 percent. A recent NLO calculation, which includes a model describing the fragmentation of the whole vector-meson nonet, describes the data within uncertainties below $6$ GeV/c, while it overestimates the data by up to 50 percent for higher $ p_{\text{T}}$.  The $\omega/\pi^0$ ratio is in agreement with previous measurements at lower collision energies and the PYTHIA calculations.  In addition, the measurement is compatible with transverse mass scaling within the measured $p_{\text{T}}$~range and the ratio is constant with $C^{\omega/\pi^{0}}= 0.67  \pm 0.03 \text{(stat)} \pm 0.04 \text{(sys)}$ above a transverse momentum of $2.5$ GeV/$c$.