Is there a greedy algorithm to solve the assignment problem?
The answer of your post question (already given in Yuval comment) is that there is no greedy techniques providing you the optimal answer to an assignment problem. 

The commonly used solution is the Hungarian algorithm, see 

Harold W. Kuhn, "The Hungarian Method for the assignment problem", Naval Research Logistics Quarterly, 2: 83–97, 1955

for the original paper. 

Otherwise your solution seems correct to me, and as usually with greedy algorithms, it will provide you a feasible solution which you may hope to not be "too far" from the global optimal solution.