To make a "magical attack", you have to meet one of three conditions laid out by the DMG:
... a magical attack is an attack delivered by a spell, a magic item, or another magical source
A fighter using an improvised weapon would not be a spell, so thats one condition we can cross off.
A improvised item is potentially a magic item, but it is not necessarily a magic weapon. Jeremy Crawford has stated:
A magic shield is not a magic weapon, unless its text says otherwise.
The PHB describes an improvised weapon as the following:
Sometimes characters don’t have their Weapons and have to Attack with whatever is at hand. An Improvised Weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead Goblin.
Often, an Improvised Weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the GM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her Proficiency Bonus.
Being considered magical is not the same as being a magical weapon. A magic item does not become a magical improvised weapon when used in that manner.. This particular question has been asked many times.
For the last condition, we'd have to analyze the fighter's kit to see if they are a "magical source". I've looked through most of the archetypes and do not see that any of them grant the ability to turn a mundane object into a magic weapon.
Note: I've specifically had to look up the Sage Advice to see if a Eldritch Knight's bonded weapon was magic, and it isn't. Nothing else would come close to achieving this.
So to summarize, it does not appear a fighter has any way to make a magical attack with an improvised weapon..