I have a texture of the earth which I want to map onto a sphere.
As it is a unit sphere and the model itself has no texture coordinates, the easiest thing I could think of is to just calculate spherical coordinates for each vertex and use them as texture coordinates.
textureCoordinatesVarying = vec2(atan(modelPositionVarying.y, modelPositionVarying.x)/(2*M_PI)+.5, acos(modelPositionVarying.z/sqrt(length(modelPositionVarying.xyz)))/M_PI);
This is a common problem. The seams between two texture coordinate topologies, where you want the texture coordinate to seamlessly wrap from 1.0 to 0.0 requires the mesh to properly handle this. To do this, the mesh must duplicate every vertex along the seam. One of the vertices will have a 0.0 texture coordinate and will be connected to the vertices coming from the right (in your example). The other will have a 1.0 texture coordinate and will be connected to the vertices coming from the left (in your example).
This is a mesh problem, and it is best to solve it in the mesh itself. The same position needs two different texture coordinates, so you must duplicate the position in question.
Alternatively, you could have the fragment shader generate the texture coordinate from an interpolated vertex normal. Of course, this is more computationally expensive, as it requires doing a conversion from a direction to a pair of angles (and then to the [0, 1] texture coordinate range).