I want to move the selected vertex along the blue axis until it exactly meets the intersecting edge. How can I make sure it exactly touches the edge without gaps or overlap?
Hold CTRL while moving the vertex and it will snap to the midpoint of the edge (it will also snap to other vertices). Once you get it on the edge, you can switch to the "Normal" coordinate system and slide it back and forth along the edge.
Dynamic Context Menu
Enable Dynamic Context Menu from UserPreferences > Add-ons
then select the two edges that intersect
then hit Spacebar > Snap Cursor > Cursor to Edge Intersection
that moves the 3d cursor to the intersection
then pick the vertex you want to move to that position, then pick Mesh > Snap > Selection to cursor.
then you get this.
If you need to repeat this kind of operation many times during your session then get acquainted with the keyboard shortcuts whenever they're listed beside the menu entries.
In Edit Mode:
First press Ctrl+Shift+TAB and select Edge to snap to Edges.
Then press G and Z to move along the blue axis (or use manipulator) and aim the mouse at the part of the edge you want to snap to, press and hold Ctrl to enable snapping and then press LMB or Enter whenever you're satisfied with the position.
This makes the vertex go along the Z axis to align with the edge you hovered over while holding Ctrl in 3D view.