I had to do this recently but unfortunately I wasn't using a simple vertex/index format so the details where somewhat different.  Going from how I did it and from the top of my head, I would say you do something like as follows........ Find the  two (or possibly one triangle, if it's on the edge of the mesh) that contain the two vertexes you are going to merge (i.e. they share and edge). These two triangles are simply removed and the triangle list is compacted accordingly. Now you are gong to remove one of the two vertexes that are close to each other and assign any indexes in triangles that reference that vertex to the other vertex. In reality I guess you could just leave the vertex you are destroying in the list and just dereference it, or you could actually remove it and compact the vertex list.  Hope I'm steering you in the right direction.

The simplest way is to just reconstruct a new mesh:

for each triangle of original mesh

{

create new triangle in new mesh

for each original triangle vertex

{

if new mesh already has a similar vertex use its index; else create new vertex and use its index

set vertex index to the new triangle 

}

}

Edit: An entire new mesh is not necessary, but an old and a growing new vertex buffer have to exist while the algorithm is running.

This is how I do it in my Mesh class:

        /// <summary>
        /// Welds close vertices into one.
        /// </summary>
        private void weldVertices( int[] tvis_ )
        {
            const float sqrDistThreshold = DupedVertexMinDistance * DupedVertexMinDistance;
            var duplicatedVertices = new Dictionary<int, int>();
            for( var i = 0; i < vertexCount - 1; ++i )
                for( var j = i + 1; j < vertexCount; ++j )
                    if( !duplicatedVertices.ContainsKey( j ) )
                        if( (pointByIdx( i ) - pointByIdx( j )).sqrMagnitude <= sqrDistThreshold )
                            duplicatedVertices.Add( j, i );

            // adjust triangles
            for( var n = 0; n < indexCount; ++n )
                if( duplicatedVertices.ContainsKey( tvis_[ n ] ) )
                    tvis_[ n ] = duplicatedVertices[ tvis_[ n ] ];
        }

For performance reasons, this method just identifies duplicated vertices and corrects triangles. I later use a different method to remove orphan vertices and invalid triangles (triangles with 2 or 3 duplicated indices).