I have not been able to found any in-built way to lock the rotation of 3D graphics around one axis, so we will have to build our own solution. Let's use EventHandler to catch the movements of mouse, and then dynamically change the point of view via ViewVector and ViewVertical. We will use the natural tilt of the Earth, i.e. $23.4^\circ$.
To make the rotation more realistic, we will use the component of the mouse movement parallel to the equator. Rotation speed can be tuned with the parameter rotationSpeed, however, some additional calibration is needed if you want to make the globe rotate exactly together with the cursor. And to make everything more funky, let's use 🌍 as a mouse cursor (you can also use "LinkHand" or any other cursor you prefer).
(* Axis tilt relative to the z-axis *) tilt = 23.4 Degree; (* Rotation speed *) rotationSpeed= 3; (* Camera distance from the center *) viewDistance = 3; axis = {0, -Sin[tilt], 1}; equator = {Cos[-tilt], Sin[-tilt]}; DynamicModule[{vp = RotationTransform[0, {0, 0, 1}]@{viewDistance, 0, 0}, vv = RotationTransform[0, {0, 0, 1}]@axis, click = {0, 0}, lastPhi = 0, phi = 0}, EventHandler[ MouseAppearance[ SphericalPlot3D[1, {u, 0, Pi}, {v, 0, 2 Pi}, PlotPoints -> 50, MaxRecursion -> 0, TextureCoordinateFunction -> ({#5, 1 - #4} &), Boxed -> False, Method -> None, Axes -> False, SphericalRegion -> Sphere[{0, 0, 0}, 1.2], Mesh -> True, ViewPoint -> Dynamic[vp], ViewVertical -> Dynamic[vv], PlotStyle -> Directive[Texture[Import["https://i.sstatic.net/5JpK4.jpg"]], Specularity[White, 1000]], Lighting -> "Neutral"], Graphics[Style[Text@"\|01f30d ", 40]]], { "MouseDown" :> (click = MousePosition["Graphics"]), "MouseDragged" :> ( phi = rotationSpeed*(equator . (click - MousePosition["Graphics"])); rot = RotationTransform[lastPhi + phi, {0, 0, 1}]; vp = rot[{viewDistance, 0, 0}] ; vv = rot[axis]), "MouseUp" :> (lastPhi = lastPhi + phi) }]] 
Old Answer
I misunderstood initially what the question is really about, so I have provided a new answer above.
Firstly, the option is actually documented under Graphics3D (Options > Method > RotationControl). Secondly, you have a bug in your code: comma instead of an arrow in "RotationControl" -> "Globe". Thirdly, use "RotationMode" -> "SphericalRegion" to stabilize rotation.
