The force:recordData component also provides a delete function, for the ability to delete records. Note that you will need to have recordId populated in order to delete the record.

The component code is as follows:

<aura:component implements="flexipage:availableForRecordHome,force:hasRecordId">

    <aura:attribute name="recordError" type="String" access="private"/>
  
    <force:recordData aura:id="recordHandler"
        recordId="{!v.recordId}"
        fields="Id"
        targetError="{!v.recordError}"
        recordUpdated="{!c.handleRecordUpdated}" />

    <!-- Display Lightning Data Service errors, if any -->
    <aura:if isTrue="{!not(empty(v.recordError))}">
        <div class="recordError">
            <ui:message title="Error" severity="error" closable="true">
                {!v.recordError}
            </ui:message>
        </div>
    </aura:if>

    <div class="slds-form-element">
        <Lightning:button
            label="Delete Record"
            onclick="{!c.handleDeleteRecord}"
            variant="brand" />
    </div>
</aura:component>

The controller code snippet with the deleteRecord() function is as follows:

({
    handleDeleteRecord: function(component, event, helper) {
        component.find("recordHandler").deleteRecord($A.getCallback(function(deleteResult) {
            // NOTE: If you want a specific behavior(an action or UI behavior) when this action is successful 
            // then handle that in a callback (generic logic when record is changed should be handled in recordUpdated event handler)
            if (deleteResult.state === "SUCCESS" || deleteResult.state === "DRAFT") {
                // record is deleted
                console.log("Record is deleted.");
            } else if (deleteResult.state === "INCOMPLETE") {
                console.log("User is offline, device doesn't support drafts.");
            } else if (deleteResult.state === "ERROR") {
                console.log('Problem deleting record, error: ' + JSON.stringify(deleteResult.error));
            } else {
                console.log('Unknown problem, state: ' + deleteResult.state + ', error: ' + JSON.stringify(deleteResult.error));
            }
        }));
    },

    /**
     * Control the component behavior here when record is changed (via any component)
     */
    handleRecordUpdated: function(component, event, helper) {
        var eventParams = event.getParams();
        if(eventParams.changeType === "CHANGED") {
           // record is changed
        } else if(eventParams.changeType === "LOADED") {
            // record is loaded in the cache
        } else if(eventParams.changeType === "REMOVED") {
            // record is deleted, show a toast UI message
            var resultsToast = $A.get("e.force:showToast");
            resultsToast.setParams({
                "title": "Deleted",
                "message": "The record was deleted."
            });
            resultsToast.fire();

        } else if(eventParams.changeType === "ERROR") {
            // there's an error while loading, saving, or deleting the record
        }
    }
})

Notice that upon successful deletion, the changeType in the event parameters handled on the record update will include the REMOVED keyword.