The idea that has been floated is that ignoring gravitomagnetism is a mistake: that the anomalous rotation curves of galaxies is, in fact, a result of this unwarranted approximation, and that ones this important general relativistic effect is accounted for, dark matter is no longer needed to explain what we see.

I am deeply skeptical of these claims for several reasons, including the fact that it is very difficult to reconcile a universe with no dark matter and no modified gravity with cosmological data, such as the statistical distribution of matter or the angular power spectrum of the cosmic microwave background.

But I also have a more direct concern: the magnitudes just don't add up. Gravitomagnetism is a second-order, "post-Newtonian" effect, and that remains true even on galactic scales. Galaxies are big things, to be sure, but not nearly big enough compared to the size of the visible universe.

But enough philosophizing. Let's look at numbers.

Gravitomagnetism arises from a simple approximation: a solution of Einstein's field equations in the presence of slowly moving matter. This solution can be written in a very general form, valid to ${\cal O}(c^{-4}),$ as

$$ds^2=\left(1+2\frac{\Phi}{c^2}\right)c^2dt^2-2\left(\frac{{\bf A}}{c}\cdot d{\bf r}\right)c~dt-\left(1-2\frac{\Phi}{c^2}\right)dr^2,\tag{1}$$

where $\Phi$ is the Newtonian gravitational potential and ${\bf A}$ is the gravitomagnetic vector potential. This form of the metric works for any perfect fluid matter distribution so long as it is nonrelativistic and slowly moving. For a compact source of mass $M$ and angular momentum ${\bf J}$ at the origin, however, we have the specific form

\begin{align}\Phi&=-\frac{GM}{r},\\~\\ {\bf A}&=-\frac{2G}{c^2}\frac{{\bf J}\times{\bf r}}{r^3}.\end{align}\tag{2}

The gravitomagnetic vector potential that corresponds to ${\bf A}$ is defined as

$${\bf B}=\nabla\times {\bf A}.\tag{3}$$

To complete the analogy with electromagnetism, we can also define the gravitoelectric vector potential in the form

$${\bf E}=\nabla\Phi-\partial_t{\bf A}.\tag{4}$$

With these definitions at hand, we can write down the gravitoelectromagnetic equivalent of the Lorentz-force, acting on a particle with mass $m,$ moving with velocity ${\bf v}$:

$${\bf F}=-m{\bf E}-m{\bf v}\times{\bf B}.\tag{5}$$

In the case of a static gravitational field and circular motion, the magnitude of the radial acceleration due to the gravitomagnetic term can be estimated easily:

$$a_B=-v|{\bf B}|=-\frac{4G}{c^2}\frac{J~v}{r^3}.\tag{6}$$

So how does this work out here in the Milky Way? To get a crude estimate of the Milky Way's angular momentum, we can assume $M_\star=10^{11}M_\odot$ and a characteristic radius of $r=8~{\rm kpc}$, which happens to be the approximate distance of the Sun from the central bulge. The orbital speed of the solar system is $\sim 200~{\rm km}/{\rm s}$. Multiplying these together, we obtain

$$J_\star=M_\star vr\sim 10^{67}~{\rm J}\cdot{\rm s}.\tag{7}$$

Correspondingly, $B\sim 2\times 10^{-21}~s^{-1}$, and the resulting radial acceleration is

$$a_B=v~B\sim 4\times 10^{-16}~{\rm m}/{\rm s}^2.\tag{8}$$

In contrast, the centrifugal acceleration of the solar system as it orbits the Milky Way is given by

$$a_\odot=\frac{v^2}{r}\sim 1.6\times 10^{-10}~{\rm m}/{\rm s}^2.\tag{9}$$

We can see, then, that the gravitomagnetic effect is quite negligible, as it is approximately six orders of magnitude smaller than the Newtonian value of the centrifugal force corresponding to the orbit of the solar system.

There is one additional possible point that might be raised. In Eq. (1) we implicitly assumed that the metric is asymptotically flat, approaching the Minkowski metric at great distances. This is obviously not true in our expanding universe. But can the effects of its deviation from a Minkowski background be significant on the scale of a single galaxy?

To answer this question, consider the size of the visible universe, characterized by its comoving radius of approximately 15 Gpc. We know that on these scales, deviations from the Minkowski background approach the scale of unity, i.e., $2\Phi/c^2\sim -1$. But we are not on gigaparsec scales. Rather, even for large galaxies the scales are measured at most in the tens of kiloparsecs. Therefore, it is reasonable to conclude that the error introduced by assuming an asymptotically Minkowskian metric will be of ${\cal O}(10^{-5})$ or less, thus negligible.

References:

T. Padmanabham, Gravitation, Cambridge U. Press (2010)
B. Mashhoon, Gravitoelectromagnetism, in: L. Iorio (ed.), The Measurement of Gravitomagnetism, Nova Science (2007)
H. C. Ohanian and R. Ruffini, Gravitation and Spacetime (2nd ed.), W. W. Norton & Co. (1994)
S. Weinberg, Gravitation and Cosmology, J. Wiley & Sons (1972)