Although The question already has an answer I feel like I should contribute some complementary information.
As you already pointed out right-handed neutrinos do not interact via the usual gauge interactions i.e. electromagnetic, strong and weak forces. That's why they are often called "sterile". And as Bert Barrois pointed out interactions via virtual Higgs are possible but in practice hard to detect.
However, there are experiments that search for right-handed sterile neutrinos. Those that come to my mind are experiments investigating neutrino oscillations. During the investigation of the oscillation properties of neutrinos produced at nuclear reactors the so-called Reactor Antineutrino Anomaly (RAA) has been observed i.e. the measured neutrino flux in short distance to the reactors was lower than expected (if one assumes the usual oscillation of neutrinos between three flavors). Those "missing" neutrinos can be explained by the oscillation of reactor neutrinos into a fourth neutrino type. For more detail on the RAA see e.g. this link .
The red line in this graph shows the expected data for the usual 3 flavor oscillations.
As you see the tendency of the neutrino events close to the reactor is to be lower than expected. However, this anomaly could in principle still be statistical. Experiments that want to further analyze this with respect to the possible existence of right-handed neutrinos are e.g. the STEREO experiment.
That being said I want to add something regarding the comments about Majorana Neutrinos. The statement that neutrinos being Majorana neutrinos would rule out the existence of right-handed neutrinos is wrong.
Let me explain this.
In the standard model Lagrangian a Dirac particle $\Psi$ consisting of two chiral fields $\Psi_R$ and $\Psi_L$ has a mass term which looks like
where m is generated by the coupling to the Higgs and its non-zero vacuum expectation value (VEV). That's why for neutrinos being Dirac particles we need a right-handed neutrino to form such a mass term.
The reason that we need the Higgs to explain the existence of fermion masses is that such a mass term needs to be a single with respect to the standard model gauge group SU3 x SU2$_L$ x U1$_Y$. This basically means the overall charges of the term need to be zero if you add up all fields inside. That, by the way, is the reason why we need the Higgs field. The weak force (represented by the SU2$_L$ group) only couples to left-handed particles. Therefore right-handed fields are not charged under it while left-handed fields are. This is solved by using the Higgs field to compensate the charge of the left-handed Field and therefore allow for mass terms to existing.
Now let's have a look at Majorana particles. Majorana particles are their own antiparticles
The interesting thing is that this particle-antiparticle conjugation $C$ flips the chirality of a field
Hence we can use this to form a Majorana mass term which looks like
not that one can do the same for right-handed fields $\Psi_R$
Now again due to left-handed fields being equally charged such a team cannot occur. (It could if we impose another second Higgs field to compensate again...)
However, as you pointed out already right-handed neutrinos are not charged under ANY of the standard model gauge group. Hence nothing forbids us to write down such a Majorana mass term for right-handed neutrinos. This means that NATURALLY right-handed neutrinos ARE MAJORANA particles. This model is well established as the seesaw type 1 model.