This ruling raises action economy, and other, problems.
For the action economy problem, you are giving the PC an additional attack that uses its Movement. Movement is not attack in the D&D 5e action economy, it is its own separate thing that is separate from attacks. The ruling that you propose significantly changes the action economy built into the game. I have outlined the action economy here.
To repeat for emphasis: in the D&D 5e action economy1 a move is not an attack. That's the rule you are suggesting be modified.
Players being subject to this option are too easily killed
Do you want your Medium-sized dwarf to be killed by a Huge creature, like a Frost Giant, who simply falls on the Dwarf after it has attacked something else with its two axe attacks? Probably not.
- To illustrate the problem that this creates: you and your Paladin ally engage a Frost Giant in melee combat. If we used your "two size" differential as a guideline, the Giant could make two attacks on your Paladin ally and then just fall on you, the Dwarf, and crush you to death. That is what could happen if you change the action economy like that.
Deal with the stirge on the next turn, and don't stretch the action economy that hard.
Can the DM rule that you can try? Sure.
But avoid an "automatic" win. The automatic win opens the door for the giants killing your dwarf by falling on him. For a lot of things in this game, a saving throw is offered when something happens. Provide the stirge a saving throw (Dex makes sense) to account for the chances that the dwarf falls awkwardly, or the stirge squirms into a nook or crevice and softens the blow. Failed save = Splat.
Can the Dwarf make a second attack if he has not used a bonus action?
There might be a way around this that uses the action economy, if the dwarf is not wielding a shield (which takes an Action to doff (p. 64, Basic Rules)).
In Two Weapon Fighting (general case) if you have a light weapon in one hand, then you can use your second hand to make an attack with a bonus action. You don't get the damage bonus from your str/dex, but an attack on the stirge can be made.
You could thus take the position that, if the Dwarf was not wielding a shield and had no other weapons to hand, that a bonus attack a la two weapon fighting is permitted on the second stirge. It's not hard to argue that an attack with "unarmed strike" less difficult than an attack with a light weapon.
Two-Weapon Fighting (Basic Rules, p. 74)
When you take the Attack action and attack with a light melee weapon
that you’re holding in one hand, you can use a bonus action to attack
with a different light melee weapon that you’re holding in the other
hand. You don’t add your ability modifier to the damage of the bonus
attack, unless that modifier is negative. If either weapon has the
thrown property, you can throw the weapon, instead of making a melee
attack with it.
If the Dwarf is a Ranger, Monk, Paladin, Barbarian, or Fighter of 5th level or higher, or a Valor Bard of 6th level or higher, there is already a second attack available to deal with the stirge.
If the Dwarf is a Fighter, Action Surge, if not already used, will also allow for another attack/stirge removal option.
1 Action Economy: the five elements of the action economy are Move (not an attack), Action (can be an attack), Reaction (might be an attack), Bonus Action (might be an attack) Interact with an object (not an attack). This is all found in the PHB, Chapter 9.