Dijsktra's algorithm applied to travelling salesman problem
Dijkstra's algorithm returns a shortest path tree, containing the shortest path from a starting vertex to each other vertex, but not necessarily the shortest paths between the other vertices, or a shortest route that visits all the vertices.

Here's a counter example where the greedy algorithm you describe will not work:



Starting from $a$, the greedy algorithm will choose the route $[a,b,c,d,a]$, but the shortest route starting and ending at $a$ is $[a,b,d,c,a]$.
Since the TSP route is not allowed to repeat vertices, once the greedy algorithm chooses $a,b,c,d$, it is forced to take the longest edge $d,a$ to return to the starting city.